Browse Kinematics

The bus passes $A$ with speed $3\ \text{m s}^{-1}$
The bus moves with constant acceleration from $A$ to $B$ , reaching $B$ with speed $V\ \text{m s}^{-1}$
The bus then moves with constant deceleration from $B$ until it comes to rest at $C$

The bus is modelled as a particle.
(a) Sketch a speed-time graph for the motion of the bus from

A

C

.

(3 marks)

The distance

AB

165

m and the distance

BC

120

m.
(b) Find, in terms of

V

,

(i) the time taken by the bus to travel from

A

B

(ii) the time taken by the bus to travel from

B

C

(4 marks)

The bus takes

42

s to travel from

A

C

.
(c) Find the value of

V

.

(2 marks)
(d) Find the deceleration of the bus as it moves from

B

C

.

(2 marks)

●●●●●Level 411 marksStart

A car moves along a straight horizontal road. At time

t = 0

, the car passes the point

O

with speed

5\ \text{m s}^{-1}

Figure 1 (not accurately drawn)

Figure 1 shows the acceleration-time graph for the motion of the car in the interval

0 \leqslant t \leqslant 35

, where

t

seconds is the time after the car passes

O

. The acceleration of the car is

$0.8\ \text{m s}^{-2}$ for $0 < t < 10$
$-0.2\ \text{m s}^{-2}$ for $10 < t < 25$
$-1\ \text{m s}^{-2}$ for $25 < t < 35$

The car is modelled as a particle.
(a) Show that the speed of the car when

t = 25

10\ \text{m s}^{-1}

(3 marks)
(b) Sketch a speed-time graph for the motion of the car in the interval

0 \leqslant t \leqslant 35

(3 marks)
(c) Find the total distance travelled by the car in the interval

0 \leqslant t \leqslant 35

(4 marks)

●●●●●Level 410 marksStart

A small stone is projected vertically upwards with speed

16.8\ \text{m s}^{-1}

from a point

A

at the top of a vertical cliff. The point

A

38.5

m vertically above the point

B

. The point

B

lies on horizontal ground at the foot of the cliff.

The stone moves freely under gravity until it hits the ground at

B

with speed

V\ \text{m s}^{-1}

. After hitting the ground the stone does not rebound.
(a) Find the value of

V

.

(3 marks)
(b) Find the time taken for the stone to reach

B

.

(3 marks)

The point

C

9.8

m vertically above

A

.
(c) Find the length of time for which the stone is above

C

.

(4 marks)
(d) Sketch a speed-time graph for the motion of the stone from the instant it is projected to the instant it reaches

B

.

(No further calculations are required.)

(2 marks)

●●●●●Level 412 marksStart

A particle

P

moves along a straight line. At time

t

seconds, where

t \geqslant 0

, the velocity of

P

v

m s⁻¹, where

v = 3t^2 - 12t + 9

t = 0

P

is at the point

O

on the line.
(a) Find the values of

t

at which

P

is instantaneously at rest. (2 marks)
(b) Find the total distance travelled by

P

in the first

4

seconds. (6 marks)
(c) Find the magnitude of the acceleration of

P

when

t = 4

. (2 marks)

●●●●●Level 410 marksStart

Figure 1 — not accurately drawn

Figure 1 shows a speed-time graph for a model of the motion of a cyclist riding a

1000

m time trial in

75

s.

The cyclist

starts from rest at time $t = 0$ and accelerates at a constant rate, reaching a speed of $15\ \text{m s}^{-1}$ when $t = 10$
then rides at a constant speed of $15\ \text{m s}^{-1}$ from $t = 10$ until $t = 55$
then decelerates at a constant rate from $t = 55$ to $t = 75$ , crossing the finish line with speed $U\ \text{m s}^{-1}$

Using the model,
(a) find the acceleration of the cyclist during the first

10

s of the ride, stating the units of your answer,

(2 marks)
(b) find the distance covered by the cyclist during the first

55

s of the ride,

(3 marks)
(c) find the value of

U

.

(3 marks)

●●●●●Level 48 marksStart

A particle

P

moves in a straight line. The velocity

v\text{ m s}^{-1}

P

at time

t

seconds is given by

v = \tfrac{1}{16}kt(t - 4) \quad \text{for } 0 \leqslant t \leqslant 6,

v = \frac{27k}{t^2} \quad \text{for } 6 \leqslant t \leqslant 10,

where

k

is a positive constant.
(a) Describe the velocity–time graph for

P

for values of

t

from 0 to 10. You should state the values of

t

at which the graph meets the

t

-axis and the values of

v

, in terms of

k

, when

t = 6

and when

t = 10

, and describe the shape of each section of the graph.

(3 marks)
(b) Find the value of

t

, where

0 \leqslant t \leqslant 10

, at which

P

has zero acceleration.

(1 mark)
(c) In this question you must show detailed reasoning.

It is given that

P

travels a total distance of

94

m between

t = 0

and

t = 10

.

Find the value of

k

.

(6 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A particle

P

moves along a straight line. The speed of

P

at time

t

seconds is

v

m s

^{-1}

, where

v

is the continuous function

v = \begin{cases} t^2 + 3t + 2 & 0 \leqslant t \leqslant k \\ 6t + 12 & k < t \leqslant 10 \end{cases}

and

k

is a constant.
(a) Show that

k = 5

, explaining your method carefully.

(3 marks)
(b) Find the acceleration of

P

when

t = 2.5

(3 marks)

When

t = 0

P

passes through the fixed point

O

on the line.
(c) Find the distance of

P

from

O

when

t = 8

(7 marks)

●●●●●Level 513 marksStart

The points

A

and

B

lie on a straight horizontal track.Figure 1 (not accurately drawn)

A go-kart

P

starts from rest at

A

at time

t = 0

and moves along the track towards

B

. Figure 1 shows the speed-time graph for the motion of

P

.

Go-kart

P

accelerates uniformly for

10

seconds until its speed is

V\ \text{m s}^{-1}

. It then moves at constant speed

V\ \text{m s}^{-1}

. When

t = 30

P

passes

B

.

Given that the distance

AB

300

m,
(a) show that

V = 12

(3 marks)
(b) Find the acceleration of

P

between

t = 0

and

t = 10

(2 marks)

Go-kart

P

continues to move along the track in the same direction at the same constant speed,

V\ \text{m s}^{-1}

When

t = 8

, a second go-kart

Q

starts from rest at

A

and moves along the track in the same direction as

P

. Go-kart

Q

accelerates uniformly for

T

seconds until its speed is

18\ \text{m s}^{-1}

. It then moves at constant speed

18\ \text{m s}^{-1}

Go-kart

Q

catches up with

P

when

t = 32

The go-karts are modelled as particles.
(c) On a copy of Figure 1, sketch a speed-time graph showing the motion of both go-karts for the interval

0 \leqslant t \leqslant 32

(3 marks)
(d) Find the value of

T

(5 marks)

●●●●●Level 513 marksStart

Constant acceleration (suvat)

54 questions

LessonNot started

SECTION B: MECHANICS

A cyclist travels along a straight horizontal road with constant acceleration. She passes point

A

with speed

8

m s⁻¹ and,

100

m further along the road, passes point

B

with speed

18

m s⁻¹.
(a) Find the acceleration of the cyclist. (3 marks)
(b) Find the time taken to travel from

A

B

. (2 marks)
(c) State one assumption of the model used in your calculations. (1 mark)

●●●●●Level 26 marksStart

A particle moves with constant acceleration. It has initial velocity

2\ \text{m s}^{-1}

and acceleration

3\ \text{m s}^{-2}

. Find its velocity after

4

●●●●●Level 22 marksStart

A particle starts from rest and accelerates at

2\ \text{m s}^{-2}

. Find the distance travelled in

5

●●●●●Level 22 marksStart

A delivery van is initially at rest at a set of traffic lights on a straight horizontal road.

The van then moves off along the road with a constant acceleration of

1.6\ \text{m s}^{-2}

Find
(a) the speed of the van after

7

s,

(1 mark)
(b) the distance travelled by the van in the first

7

s.

(2 marks)

●●●●●Level 23 marksStart

A cyclist is initially at rest on a straight horizontal road.

The cyclist then accelerates along the road with a constant acceleration of

0.9\ \text{m s}^{-2}

Find
(a) the speed of the cyclist after

12

s,

(1 mark)
(b) the distance travelled by the cyclist in the first

12

s.

(2 marks)

●●●●●Level 23 marksStart

A particle

P

moves with constant acceleration

(1.5\mathbf{i} - 4\mathbf{j})\ \text{m s}^{-2}

. At time

t = 0

seconds,

P

is moving with velocity

(-2\mathbf{i} + 7\mathbf{j})\ \text{m s}^{-1}

.
(a) Determine the speed of

P

when

t = 4

.

(4 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the displacement of

P

from its position at

t = 0

to its position at

t = 4

.

(2 marks)

●●●●●Level 36 marksStart

A train travels between two stations

2000

m apart. It starts from rest and accelerates uniformly at

0.8

m s⁻² for

25

s, then travels at constant speed for

T

seconds, and finally decelerates uniformly at

1.25

m s⁻² to rest at the second station.
(a) Sketch a velocity–time graph for the journey, labelling the key values. (2 marks)
(b) Find the value of

T

. (4 marks)
(c) Find the average speed of the train for the whole journey. (2 marks)
(d) Explain why the average speed is less than the constant speed reached. (1 mark)

●●●●●Level 39 marksStart

SECTION B: MECHANICS

A train travels between two stations. It starts from rest and accelerates uniformly to a speed of

30

m s⁻¹ in

20

seconds. It then travels at

30

m s⁻¹ for

T

seconds, before decelerating uniformly to rest in a further

15

seconds.

The total distance between the stations is

2400

m.
(a) Find the total distance travelled during the acceleration and deceleration phases. (2 marks)
(b) Find the value of

T

. (2 marks)
(c) Find the average speed of the train for the whole journey, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 36 marksStart

A remote-controlled car moves in a straight line along a horizontal track.

The graph below shows how the car's velocity,

v\ \text{m s}^{-1}

, changes with time,

t

seconds.Over the period

0 \le t \le 20

the car has a total displacement of

-30

metres.

Initially the car has velocity

0\ \text{m s}^{-1}

Find the next time when the velocity of the car is

0\ \text{m s}^{-1}

(4 marks)

●●●●●Level 34 marksStart

A motorcyclist has an initial velocity of

2\ \text{m s}^{-1}

The motorcyclist is moving in a straight line along a horizontal road.

The acceleration,

a\ \text{m s}^{-2}

, of the motorcyclist at time

t

seconds is given by

a = 1.2 - 4kt + 9kt^2

where

k

is a constant.

When

t = 5

the motorcyclist has a velocity of

21\ \text{m s}^{-1}

Show that

k = \dfrac{1}{25}

(4 marks)

●●●●●Level 34 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

Two drops of rainwater hang, at rest, from the edges of two different balconies on the same side of a block of flats. The ground below the balconies is horizontal.

The vertical distance between the two drops is

d

metres.

At the same instant both drops begin to fall freely under gravity.

The first drop hits the ground after

1.6

seconds.

The second drop hits the ground

0.3

seconds later.

Show that

d

is approximately

5.1

(4 marks)

●●●●●Level 34 marksStart

A particle accelerates uniformly from

5\ \text{m s}^{-1}

15\ \text{m s}^{-1}

with acceleration

2\ \text{m s}^{-2}

. Find the distance travelled.

●●●●●Level 32 marksStart

A particle has initial velocity

10\ \text{m s}^{-1}

and acceleration

-2\ \text{m s}^{-2}

. Find its displacement after

3

●●●●●Level 32 marksStart

A car accelerates uniformly from

4\ \text{m s}^{-1}

20\ \text{m s}^{-1}

8

s. Find its acceleration.

●●●●●Level 32 marksStart

A particle starts at

8\ \text{m s}^{-1}

and travels

100

m in

10

s under constant acceleration. Find the acceleration.

●●●●●Level 32 marksStart

A train travelling at

30\ \text{m s}^{-1}

brakes uniformly and stops in

600

m. Find the deceleration.

●●●●●Level 32 marksStart

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$ .]

A particle

P

is moving on a smooth horizontal plane.

The particle has constant acceleration

(2\mathbf{i} + 3\mathbf{j})\ \text{m s}^{-2}

At time

t = 0

P

passes through the point

A

.

At time

t = 3

P

passes through the point

B

.

The velocity of

P

as it passes through

A

(5\mathbf{i} - 11\mathbf{j})\ \text{m s}^{-1}

(a) Find the speed of

P

as it passes through

B

.

(4 marks)

The position vector of

A

(-9\mathbf{i} - 20\mathbf{j})

m.

At time

t = T

seconds, where

T > 3

P

passes through the point

C

.

The position vector of

C

(41\mathbf{i} + c\mathbf{j})

m.
(b) Find the value of

T

.

(3 marks)
(c) Find the value of

c

.

(3 marks)

●●●●●Level 410 marksStart

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

A particle

P

of mass 6 kg is at rest at the point

A

on a smooth horizontal plane.

At time

t = 0

, two forces,

\mathbf{F}_1 = (7\mathbf{i} - 4\mathbf{j})

N and

\mathbf{F}_2 = (\lambda\mathbf{i} + \mu\mathbf{j})

N, where

\lambda

and

\mu

are constants, are applied to

P

Given that

P

moves in the direction of the vector

(4\mathbf{i} + 3\mathbf{j})

(a) show that

3\lambda - 4\mu + 37 = 0

(4 marks)

At time

t = 6

seconds,

P

passes through the point

B

.

Given that

\lambda = 5

(b) find the length of

AB

.

(5 marks)

●●●●●Level 49 marksStart

The diagram shows a small block

B

, of mass

0.6

kg, on a rough horizontal surface, and a particle

P

, of mass

0.9

kg, on a smooth plane inclined at

40°

to the horizontal. The horizontal surface meets the inclined plane along an edge, at which a small smooth pulley is fixed.

B

and

P

are attached to the ends of a light inextensible string which passes over the pulley; the string is taut, the part of the string from

B

to the pulley is horizontal and the part from

P

to the pulley is parallel to a line of greatest slope of the plane.The system is released from rest. In the first

0.5

seconds of the motion

P

moves

0.35

m down the plane and

B

does not reach the pulley.
(a) Find the tension in the string during the first

0.5

seconds of the motion.

(4 marks)
(b) Determine the coefficient of friction between

B

and the horizontal surface.

(5 marks)

●●●●●Level 49 marksStart

The points

A

and

B

are the lower and upper ends, respectively, of a line of greatest slope on a rough plane inclined at an angle

\theta

to the horizontal, where

\sin\theta = \frac{5}{13}

. The distance

AB

0.75

m (see diagram).The coefficient of friction between the plane and a particle

P

0.45

. The particle

P

is projected from

A

towards

B

, up the line of greatest slope, with speed

5.5

m s

^{-1}

.

Show that the speed of

P

B

4.3

m s

^{-1}

.

(6 marks)

●●●●●Level 46 marksStart

Two cyclists,

P

and

Q

, are riding in the same direction along a straight horizontal road. The diagram shows the velocity–time graphs for their motion.At time

t = 0

seconds, cyclist

Q

, who is riding with constant speed

6\text{ m s}^{-1}

, is overtaken by cyclist

P

, whose speed is then

9\text{ m s}^{-1}

. From

t = 0

cyclist

P

slows down with constant deceleration for 40 seconds. When

t = 40

the speed of

P

5\text{ m s}^{-1}

, and

P

keeps this speed in the subsequent motion.
(a) Calculate the deceleration of

P

.

(2 marks)
(b) Determine the value of

t

when

Q

overtakes

P

.

(4 marks)

●●●●●Level 46 marksStart

Figure 1Diagram not accurately drawn.

A quad bike of mass

400

kg is towing a cart of mass

250

kg up a straight road, as shown in Figure 1. The road is inclined at an angle

\alpha

to the horizontal, where

\sin\alpha = \frac{1}{14}

The driving force produced by the engine of the quad bike is

1000

N and the quad bike and the cart move with acceleration

0.4\ \text{m s}^{-2}

The non-gravitational resistance to the motion of

the quad bike is modelled as a constant force of magnitude $3R$ newtons
the cart is modelled as a constant force of magnitude $2R$ newtons

The quad bike and the cart are modelled as particles. The tow bar between the quad bike and the cart is modelled as a light rod that is parallel to the direction of motion.

Using the model,
(a) show that

R = 57

(4 marks)
(b) find the tension in the tow bar.

(3 marks)

When the quad bike and the cart are moving at

9\ \text{m s}^{-1}

, the tow bar breaks.

Given that the non-gravitational resistance to the motion of the cart is unchanged,
(c) use the model to find the further distance moved by the cart before it first comes to rest.

(4 marks)

●●●●●Level 411 marksStart

Figure 1Diagram not accurately drawn.

A fixed rough plane is inclined at an angle

\alpha

to the horizontal, where

\tan\alpha = \frac{3}{4}

. A small smooth pulley is fixed at the top of the plane.

A particle

P

of mass

5

kg is at rest on the plane. One end of a light inextensible string is attached to

P

. The string passes over the pulley and its other end is attached to a particle

Q

of mass

4

kg, which hangs freely, vertically below the pulley, as shown in Figure 1.

The part of the string from

P

to the pulley lies along a line of greatest slope of the plane. The coefficient of friction between

P

and the plane is

\mu

.

The system is in limiting equilibrium, with the string taut and

P

on the point of slipping up the plane.
(a) Find the value of

\mu

.

(8 marks)

The string breaks and

P

begins to slide down the plane.
(b) Find the time taken for

P

to slide

2.5

m down the plane.

(4 marks)

●●●●●Level 412 marksStart

Figure 1Diagram not accurately drawn.

In a table game, two discs,

P

and

Q

, rest on a rough horizontal board. Disc

P

has mass

2m

kg and disc

Q

has mass

5m

kg. The distance between

P

and

Q

3.2

m, as shown in Figure 1. The discs are modelled as particles.

Disc

P

is struck so that it receives an impulse of magnitude

\lambda

N s in the direction

PQ

. Immediately after the impulse is applied, the speed of

P

8\text{ m s}^{-1}

(a) Find

\lambda

in terms of

m

.

(2 marks)

Immediately before

P

collides with

Q

, the speed of

P

6\text{ m s}^{-1}

The coefficient of friction between

P

and the board is

\mu

(b) Find the value of

\mu

(7 marks)

Immediately after

P

collides with

Q

, the speed of

Q

2.2\text{ m s}^{-1}

(c) Find the speed of

P

immediately after the collision.

(3 marks)

●●●●●Level 412 marksStart

Three points

A

B

and

C

lie in that order on a straight horizontal road.

A bus moves along the road from

A

C

The bus passes $A$ with speed $3\ \text{m s}^{-1}$
The bus moves with constant acceleration from $A$ to $B$ , reaching $B$ with speed $V\ \text{m s}^{-1}$
The bus then moves with constant deceleration from $B$ until it comes to rest at $C$

The bus is modelled as a particle.
(a) Sketch a speed-time graph for the motion of the bus from

A

C

.

(3 marks)

The distance

AB

165

m and the distance

BC

120

m.
(b) Find, in terms of

V

,

(i) the time taken by the bus to travel from

A

B

(ii) the time taken by the bus to travel from

B

C

(4 marks)

The bus takes

42

s to travel from

A

C

.
(c) Find the value of

V

.

(2 marks)
(d) Find the deceleration of the bus as it moves from

B

C

.

(2 marks)

●●●●●Level 411 marksStart

A car moves along a straight horizontal road. At time

t = 0

, the car passes the point

O

with speed

5\ \text{m s}^{-1}

Figure 1 (not accurately drawn)

Figure 1 shows the acceleration-time graph for the motion of the car in the interval

0 \leqslant t \leqslant 35

, where

t

seconds is the time after the car passes

O

. The acceleration of the car is

$0.8\ \text{m s}^{-2}$ for $0 < t < 10$
$-0.2\ \text{m s}^{-2}$ for $10 < t < 25$
$-1\ \text{m s}^{-2}$ for $25 < t < 35$

The car is modelled as a particle.
(a) Show that the speed of the car when

t = 25

10\ \text{m s}^{-1}

(3 marks)
(b) Sketch a speed-time graph for the motion of the car in the interval

0 \leqslant t \leqslant 35

(3 marks)
(c) Find the total distance travelled by the car in the interval

0 \leqslant t \leqslant 35

(4 marks)

●●●●●Level 410 marksStart

A lorry moves along a straight horizontal road with constant acceleration.

The lorry passes the point

A

with speed

u\ \text{m s}^{-1}

and,

16

s later, passes the point

B

with speed

22\ \text{m s}^{-1}

The distance

AB

256

m.

The lorry is modelled as a particle.
(a) Show that

u = 10

(2 marks)
(b) Find the time taken for the lorry to travel from

A

to the midpoint of

AB

.

(5 marks)

The mass of the lorry is

2400

kg. As the lorry moves from

A

B

, the resistance to the motion of the lorry is modelled as a constant force of magnitude

900

N.
(c) Find the magnitude of the driving force produced by the engine of the lorry as it moves from

A

B

.

(3 marks)

●●●●●Level 410 marksStart

Figure 1 (not accurately drawn)

A small package

P

of mass

3

kg is placed on a rough ramp. The ramp is modelled as a rough plane inclined at an angle

\alpha

to the horizontal, where

\tan\alpha = \frac{5}{12}

, and the package is modelled as a particle.

The package is held in equilibrium by a horizontal force of magnitude

H

newtons, as shown in Figure 1. The line of action of the horizontal force lies in a vertical plane containing a line of greatest slope of the ramp.

The coefficient of friction between the package and the ramp is

\frac{1}{4}

(a) Find the smallest possible value of

H

.

(6 marks)

The horizontal force is now removed and the package starts to slide down the ramp from rest.

The package slides

2

m down the ramp in the first

T

seconds after the force is removed.
(b) Find the value of

T

.

(6 marks)

●●●●●Level 412 marksStart

A particle

P

of mass

3m

and a particle

Q

of mass

km

, where

k

is a constant, are moving in opposite directions along the same straight line on a smooth horizontal surface. Particle

P

collides directly with

Q

.

Immediately before the collision the speed of

P

7u

and the speed of

Q

2u

. Immediately after the collision the speed of

P

v

and the speed of

Q

4v

. As a result of the collision

the direction of motion of $P$ is unchanged
the direction of motion of $Q$ is reversed

The coefficient of restitution between

P

and

Q

\frac23

(a) Show that

v = 2u

(3 marks)
(b) Find the value of

k

.

(3 marks)

After the collision,

Q

hits a smooth fixed vertical wall that is perpendicular to the direction of motion of

Q

, and rebounds. When

P

and

Q

first collide, they are at a distance

d

from the wall. They collide for a second time at a distance

\frac13 d

from the wall. The coefficient of restitution between

Q

and the wall is

f

.
(c) Find the value of

f

.

(5 marks)

●●●●●Level 411 marksStart

A particle

A

of mass

2

kg lies on a rough plane inclined at an angle

\alpha

to the horizontal, where

\sin\alpha = \dfrac35

. The particle is attached to one end of a light inextensible string. The string lies along a line of greatest slope of the plane and passes over a small smooth pulley

P

fixed at the top of the plane. The other end of the string is attached to a particle

B

of mass

3

kg, which hangs freely with the string vertical.

B

1

m above horizontal ground, and the plane is long enough that

A

does not reach

P

in the motion that follows.

The coefficient of friction between

A

and the plane is

\dfrac12

. The system is released from rest with the string taut. Take

g = 9.8

m s⁻².
(a) Show that the acceleration of

A

1.96

m s⁻² and find the tension in the string. (5 marks)

When

B

reaches the ground it does not rebound.
(b) Find the further distance that

A

travels up the plane after

B

reaches the ground, before

A

first comes to instantaneous rest. (3 marks)
(c) Determine whether

A

remains at rest at this point. (2 marks)

●●●●●Level 410 marksStart

A small stone is released from rest at the top of a deep well and falls freely towards the water at the bottom of the well.

The stone passes through two different fixed points,

A

and

B

, before reaching the water.

At

A

the stone has velocity

u\ \text{m s}^{-1}

B

the stone has velocity

4u\ \text{m s}^{-1}

It can be assumed that:

* the motion is due to gravitational force only * the acceleration due to gravity remains constant throughout.
(a) Show that the time taken for the stone to travel from

A

B

\dfrac{3u}{g}

seconds.

(2 marks)
(b) Point

A

h

metres above the water.

Show that

h > \frac{15u^2}{2g}

Fully justify your answer.

(3 marks)

●●●●●Level 45 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A machine for cricket catching practice fires cricket balls upwards. Each ball leaves the machine at ground level with a speed of

11.2\ \text{m s}^{-1}

The manufacturer claims that the balls can reach a maximum height of

6.4

metres above the ground.
(a) Suppose that the machine fires the balls vertically upwards.
(a) (i) Verify the manufacturer's claim.

(ii) State two modelling assumptions you have made in verifying this claim.

(4 marks)
(b) In fact the machine fires the balls in a direction anywhere between

0

and

30

degrees from the vertical.

The range of maximum heights,

h

metres, above the ground which can be reached by the balls may be expressed as

k < h \le 6.4

Find the value of

k

(4 marks)

●●●●●Level 48 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

Two sledges,

A

and

B

, are connected to each other by a light, rigid pole. Sledge

A

has a mass of

12

kilograms. Sledge

B

has a mass of

8

kilograms.

A child pulls sledge

A

using a rope inclined at an angle of

20^\circ

above the horizontal. The tension in the rope is

40

newtons.

As a result the sledges move forward, from rest, in a straight line across horizontal snow with acceleration

0.75\ \text{m s}^{-2}

as shown in the diagram below.As it moves sledge

A

experiences a total resistance force of

14

N
(a) Show that the total resistance force experienced by sledge

B

is approximately

8.6

N

(4 marks)
(b) At the instant that the sledges reach a speed of

3\ \text{m s}^{-1}

the child lets go of the rope.

The sledges decelerate at a constant rate until they come to rest. It can be assumed that the resistance forces remain unchanged.

Find the distance the sledges travel after the child lets go of the rope.

(3 marks)
(c) State one modelling assumption that you have used about the pole when answering part (a).

(1 mark)

●●●●●Level 48 marksStart

Two particles,

P

and

Q

, move in the same horizontal plane.

Particle

P

is initially at rest at the point with position vector

(2\mathbf{i} + \mathbf{j})

metres and moves with constant acceleration

(3\mathbf{i} - 2\mathbf{j})\ \text{m s}^{-2}

Particle

Q

moves in a straight line, passing through the points with position vectors

(10\mathbf{i} - 3\mathbf{j})

metres and

(\mathbf{i} + c\mathbf{j})

metres, where

c

is a constant.

P

and

Q

are moving along parallel paths.
(a) Show that

c = 3

(4 marks)
(b) (i) Find an expression for the position vector of

P

at time

t

seconds.

(ii) Hence, prove that the paths of

P

and

Q

are not collinear.

(4 marks)

●●●●●Level 48 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

Aisha is running a goalkeeping drill with Jonah on a large playing field. The playing field can be assumed to be level and horizontal.

Aisha and Jonah are initially standing next to each other.

Aisha throws a ball forward, from a height of

1.6

metres above the playing field, at an angle of

65^\circ

above the horizontal with a speed of

10\ \text{m s}^{-1}

Exactly

0.25

seconds after the ball is thrown, Jonah sets off from Aisha and runs across the playing field, in a straight line with a constant acceleration

a\ \text{m s}^{-2}

Jonah catches the ball when it is

0.4

metres above the playing field, as shown in the diagram below.Find

a

(7 marks)

●●●●●Level 47 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

Mei and Tomasz are practising catching on a sports field. The sports field can be assumed to be level and horizontal.

Mei stands at the point

M

and Tomasz stands at the point

T

, which is

5

metres in front of Mei.

Mei throws a ball forward, towards Tomasz, from a height of

1.9

metres above the sports field, at an angle of

55^\circ

above the horizontal with a speed of

9\ \text{m s}^{-1}

Exactly

0.4

seconds after the ball is thrown, Tomasz sets off from rest at

T

and runs directly away from Mei, in a straight line with a constant acceleration

a\ \text{m s}^{-2}

Tomasz catches the ball when it is

1.2

metres above the sports field, as shown in the diagram below.Find

a

(7 marks)

●●●●●Level 47 marksStart

Figure 1 — not accurately drawn

Figure 1 shows a speed-time graph for a model of the motion of a cyclist riding a

1000

m time trial in

75

s.

The cyclist

starts from rest at time $t = 0$ and accelerates at a constant rate, reaching a speed of $15\ \text{m s}^{-1}$ when $t = 10$
then rides at a constant speed of $15\ \text{m s}^{-1}$ from $t = 10$ until $t = 55$
then decelerates at a constant rate from $t = 55$ to $t = 75$ , crossing the finish line with speed $U\ \text{m s}^{-1}$

Using the model,
(a) find the acceleration of the cyclist during the first

10

s of the ride, stating the units of your answer,

(2 marks)
(b) find the distance covered by the cyclist during the first

55

s of the ride,

(3 marks)
(c) find the value of

U

.

(3 marks)

●●●●●Level 48 marksStart

Particles

A

and

B

, of masses

3

kg and

4

kg respectively, are attached to the ends of a light inextensible string.

A

is in contact with a rough plane inclined at

\theta

to the horizontal, where

\sin\theta = \frac{7}{25}

. The string is taut and passes over a small smooth pulley

P

at the top of the plane, with the part of the string from

A

P

parallel to a line of greatest slope of the plane.

B

hangs freely below

P

1.25

m above horizontal ground, as shown in the diagram.The coefficient of friction between

A

and the plane is

\mu

. The system is released from rest and in the subsequent motion

B

hits the ground before

A

reaches

P

. The speed of

B

at the instant that it hits the ground is

2.5

m s

^{-1}

.
(a) For the motion before

B

hits the ground, show that the acceleration of

B

2.5

m s

^{-2}

.

(1 mark)
(b) For the motion before

B

hits the ground, show that the tension in the string is

29.2

N.

(3 marks)
(c) Determine the value of

\mu

.

(5 marks)

After

B

hits the ground,

A

continues to move up the plane and comes to instantaneous rest before it reaches

P

.
(d) Determine the distance that

A

moves up the plane from the instant that

B

hits the ground until

A

comes to instantaneous rest.

(4 marks)

●●●●●Level 513 marksStart

A fixed wedge has two plane faces which meet along a horizontal ridge. One face is inclined at

\alpha

to the horizontal and the other at

\beta

to the horizontal, where

\sin\alpha = \frac{12}{13}

and

\sin\beta = \frac{5}{13}

. Both faces are rough.

A particle

A

of mass

6

kg on the first face and a particle

B

of mass

3

kg on the second face are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the ridge. The parts of the string from

A

to the pulley and from

B

to the pulley are parallel to lines of greatest slope of the corresponding faces (see diagram).The coefficient of friction between

A

and the first face is

12\mu

and the coefficient of friction between

B

and the second face is

5\mu

, where

\mu

is a constant.

The system is released from rest, and in the subsequent motion

A

moves down the first face and

B

does not reach the pulley.
(a) Determine, in terms of

g

, the tension in the string.

(7 marks)

When

A

is moving at

1.5

m s

^{-1}

the string breaks. In the

0.4

seconds after the string breaks

A

moves

1.15

m down the first face.
(b) Determine the deceleration of

B

after the string breaks. Give your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 512 marksStart

The points

A

and

B

are the lower and upper ends, respectively, of a line of greatest slope on a plane inclined at an angle

\theta

to the horizontal, where

\sin\theta = \frac{8}{17}

and

AB = 2.55

m (see diagram).A particle

P

is projected up the plane with speed

7

m s

^{-1}

from

A

towards

B

.

The plane at

A

is fixed to horizontal ground.

The surface of the plane is rough and the coefficient of friction between

P

and the plane is

0.3

.
(a) Show that the speed of

P

B

3.5

m s

^{-1}

.

(6 marks)

The particle leaves the plane at

B

and moves freely under gravity. It first lands at a point

C

on the horizontal ground. The time taken for

P

to travel from

A

C

T

seconds.
(b) Determine the value of

T

.

(6 marks)

●●●●●Level 512 marksStart

Axes

Ox

and

Oy

are fixed, with

Ox

horizontal along level ground and

Oy

vertically upwards. The units of

x

and

y

are metres. A small stone

S

is thrown from

O

with a velocity whose components along

Ox

and

Oy

are

U

m s

^{-1}

and

V

m s

^{-1}

respectively.

S

is modelled as a particle moving freely under gravity.
(a) Show that the path of

S

has equation

y = \frac{V}{U}x - \frac{g}{2U^2}x^2

.

(3 marks)

During its motion,

S

just clears a vertical fence of height

\frac{3}{4}a

m at a horizontal distance

a

m from

O

S

first lands on the ground at a horizontal distance

2a

m beyond the fence.
(b) Determine the angle of projection of

S

. Give your answer in degrees correct to 3 significant figures.

(5 marks)
(c) Given that the speed of projection of

S

15

m s

^{-1}

, determine the value of

a

.

(2 marks)
(d) Hence find the maximum height of

S

above the ground during its motion.

(3 marks)
(e) State one refinement of the model, other than including air resistance, that would make it more realistic.

(1 mark)

●●●●●Level 514 marksStart

A small beanbag

P

is thrown from a point

O

on horizontal ground.

P

moves in the vertical plane containing a fixed horizontal axis

Ox

, which lies along the ground, and a fixed vertical axis

Oy

P

is projected from

O

with a velocity whose components along

Ox

and

Oy

are

U

m s

^{-1}

and

V

m s

^{-1}

respectively, and is modelled as a particle moving freely under gravity.

P

first lands on the ground at the point

C

.
(a) Determine, in terms of

U

V

and

g

, the distance

OC

.

(4 marks)

On its way to

C

P

passes through two small hoops

A

and

B

, each at a height

h

m above the ground and a horizontal distance

d

m apart (see diagram).(b) Write down the horizontal and vertical components of the velocity of

P

as it passes through

A

.

(2 marks)
(c) Hence determine an expression for

d

in terms of

U

V

g

and

h

.

(3 marks)

It is given that, as

P

passes through

A

, its direction of motion is inclined to the horizontal at an angle

\alpha

, where

\tan\alpha = \frac{3}{4}

.
(d) Determine an expression, in terms of

g

d

and

h

, for the speed with which

P

is projected from

O

.

(4 marks)

●●●●●Level 513 marksStart

A van starts from rest and moves along a straight horizontal road. For the first 15 seconds of the motion, the velocity of the van,

v\text{ m s}^{-1}

, at time

t

seconds after it starts is given by

v = kt - 0.02t^2,

where

k

is a constant.

At the instant when

t = 15

the acceleration of the van is

0.9\text{ m s}^{-2}

. From

t = 15

the van moves with constant acceleration

0.9\text{ m s}^{-2}

until its speed is

22\text{ m s}^{-1}

.
(a) Find the value of

k

.

(3 marks)
(b) Find the total distance travelled by the van from the start until its speed is

22\text{ m s}^{-1}

.

(7 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A particle

P

moves along a straight line. The speed of

P

at time

t

seconds is

v

m s

^{-1}

, where

v

is the continuous function

v = \begin{cases} t^2 + 3t + 2 & 0 \leqslant t \leqslant k \\ 6t + 12 & k < t \leqslant 10 \end{cases}

and

k

is a constant.
(a) Show that

k = 5

, explaining your method carefully.

(3 marks)
(b) Find the acceleration of

P

when

t = 2.5

(3 marks)

When

t = 0

P

passes through the fixed point

O

on the line.
(c) Find the distance of

P

from

O

when

t = 8

(7 marks)

●●●●●Level 513 marksStart

Figure 2Diagram not accurately drawn.

Two small blocks,

A

and

B

, are at rest on a rough horizontal floor. Block

A

has mass

0.6

kg and block

B

has mass

0.9

kg. The distance between

A

and

B

1.25

m, as shown in Figure 2. The blocks are modelled as particles.

The coefficient of friction between each block and the floor is

\frac{2}{7}

Block

A

is hit so that it receives an impulse of magnitude

2.4

N s in the direction

AB

.
(a) Find the speed of

A

immediately after the impulse is applied.

(2 marks)
(b) Show that the speed of

A

immediately before it collides with

B

3\text{ m s}^{-1}

(5 marks)

In the collision,

A

and

B

join together and move as a single particle.
(c) Find the speed of the combined particle immediately after the collision.

(2 marks)
(d) Find the magnitude of the impulse exerted on

A

B

in the collision, stating the units of your answer.

(2 marks)
(e) Find the distance travelled by the combined particle after the collision until it comes to rest.

(3 marks)

●●●●●Level 514 marksStart

The points

A

and

B

lie on a straight horizontal track.Figure 1 (not accurately drawn)

A go-kart

P

starts from rest at

A

at time

t = 0

and moves along the track towards

B

. Figure 1 shows the speed-time graph for the motion of

P

.

Go-kart

P

accelerates uniformly for

10

seconds until its speed is

V\ \text{m s}^{-1}

. It then moves at constant speed

V\ \text{m s}^{-1}

. When

t = 30

P

passes

B

.

Given that the distance

AB

300

m,
(a) show that

V = 12

(3 marks)
(b) Find the acceleration of

P

between

t = 0

and

t = 10

(2 marks)

Go-kart

P

continues to move along the track in the same direction at the same constant speed,

V\ \text{m s}^{-1}

When

t = 8

, a second go-kart

Q

starts from rest at

A

and moves along the track in the same direction as

P

. Go-kart

Q

accelerates uniformly for

T

seconds until its speed is

18\ \text{m s}^{-1}

. It then moves at constant speed

18\ \text{m s}^{-1}

Go-kart

Q

catches up with

P

when

t = 32

The go-karts are modelled as particles.
(c) On a copy of Figure 1, sketch a speed-time graph showing the motion of both go-karts for the interval

0 \leqslant t \leqslant 32

(3 marks)
(d) Find the value of

T

(5 marks)

●●●●●Level 513 marksStart

Figure 1 (not accurately drawn)

A sledge

P

of mass

m

is pulled up a rough slope by a rope. The slope is modelled as a rough plane inclined to the horizontal at an angle

\alpha

, where

\tan\alpha = \frac{7}{24}

The sledge is modelled as a particle and the rope is modelled as a light inextensible string.

The rope is inclined to the slope at an angle

\beta

, where

\tan\beta = \frac{3}{4}

, and the tension in the rope is

\frac{4}{5}mg

, as shown in Figure 1. The rope lies in the vertical plane containing a line of greatest slope of the plane.

The coefficient of friction between the sledge and the slope is

\frac{1}{4}

The sledge starts from rest at the point

A

on the slope and moves up the line of greatest slope through

A

.
(a) Show that the acceleration of the sledge is

\frac{6}{25}g

(8 marks)

The sledge passes through the point

B

, where

AB = 3

m.
(b) Find the speed of the sledge at

B

.

(2 marks)

When the sledge reaches

B

, the rope is released. The sledge continues to move up the slope and comes to instantaneous rest at the point

C

.
(c) Find the distance

BC

.

(3 marks)
(d) Determine whether the sledge remains at rest at

C

. You must justify your reasoning.

(2 marks)

●●●●●Level 515 marksStart

Particle

P

has mass

3m

and particle

Q

has mass

m

. The particles are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of

P

5u

and the speed of

Q

3u

. The coefficient of restitution between

P

and

Q

e

.
(a) (i) Show that the speed of

Q

immediately after the collision is

3(1 + 2e)u

(ii) Find, in terms of

e

and

u

, the speed of

P

immediately after the collision.

(7 marks)

Given that

e = \frac12

(b) find, in terms of

m

and

u

, the total kinetic energy lost in the collision.

(3 marks)

After the collision,

Q

hits a smooth fixed vertical wall that is perpendicular to the direction of motion of

Q

. The coefficient of restitution between

Q

and the wall is

\frac13

. The collision between

P

and

Q

takes place at a distance

d

from the wall.
(c) Find, in terms of

d

and

u

, the time between the first collision and the second collision between

P

and

Q

.

(5 marks)

●●●●●Level 515 marksStart

Figure 3 shows a car of mass

1200

kg travelling along a straight road inclined at an angle

\theta

to the horizontal, where

\sin\theta = \dfrac{1}{14}

.Figure 3 (not accurately drawn)

The car experiences a constant resistance to motion of

400

N. Take

g = 9.8

m s⁻².
(a) The car accelerates up the slope at

0.8

m s⁻². Find the magnitude of the driving force. (5 marks)
(b) When the car is travelling up the slope at

20

m s⁻¹ the engine is switched off. The resistance remains

400

N.

Find the distance the car travels up the slope before coming to instantaneous rest, giving your answer to

3

significant figures. (5 marks)
(c) Find the time taken for the car to come to instantaneous rest after the engine is switched off, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 marksStart

Two particles

P

and

Q

, of masses

3

kg and

5

kg respectively, are connected by a light inextensible string. Particle

P

lies on a rough plane inclined at an angle

\alpha

to the horizontal, where

\tan\alpha = \dfrac{3}{4}

. The string passes over a small smooth pulley fixed at the top edge of the plane, and

Q

hangs freely below the pulley. The part of the string from

P

to the pulley lies along a line of greatest slope of the plane. The coefficient of friction between

P

and the plane is

\dfrac{1}{2}

.

Initially

P

2.5

m from the pulley and

Q

1.6

m above horizontal ground. The system is released from rest with the string taut. Take

g = 9.8

m s⁻².
(a) Show that the acceleration of

P

2.45

m s⁻² and find the tension in the string. (6 marks)
(b) Find the time taken, from release, for

Q

to reach the ground, and the speed of

Q

at that instant. (3 marks)
(c)

Q

does not rebound when it hits the ground. Find the distance

P

moves up the plane after

Q

hits the ground before

P

first comes to instantaneous rest. (3 marks)
(d) Explain why

P

does not remain at rest at this position. (1 mark)

●●●●●Level 513 marksStart

Figure 3 shows a block

A

of mass

4

kg lying on a rough horizontal table. The block is attached to one end of a light inextensible string which passes over a smooth pulley fixed at the edge of the table. The other end of the string is attached to a block

B

of mass

3

kg, which hangs freely.Figure 3 (not accurately drawn)

The coefficient of friction between

A

and the table is

0.25

. The system is released from rest with the string taut.
(a) Find the acceleration of the system and the tension in the string. (7 marks)
(b)

B

hits the ground after falling

1.5

m. Find the speed of

B

as it hits the ground, giving your answer to

3

significant figures. (3 marks)
(c) After

B

hits the ground the string becomes slack and

A

continues to move. Find the further distance

A

travels before coming to rest, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A rough plank is

5

metres long and is inclined at an angle of

20^\circ

above the horizontal. The bottom of the plank is at the point

O

.

A sack of gravel of mass

30

kg is at rest at the point

A

on the plank. The sack is pulled up the plank using a rope attached to the sack.

Once in motion, the rope remains taut and parallel to the line of greatest slope of the plank.
(a) The tension in the rope is

200

N. The sack accelerates up the plank at

0.8\ \text{m s}^{-2}

Find the coefficient of friction between the sack and the plank.

(7 marks)
(b) (i) The sack takes

3

seconds to reach the top of the plank.

Find the distance

OA

.

(ii) Other than air resistance, state one assumption you have made about the sack in answering part (b)(i).

(4 marks)

●●●●●Level 511 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A child's wooden toy comprises a tractor and a cart connected to each other by a light, inextensible rod.

The tractor has a mass of

0.6

kilograms. The cart has a mass of

0.4

kilograms.

A string inclined at an angle of

30^\circ

above the horizontal is attached to the front of the tractor. The tension in the string is

5

newtons.

As a result the toy moves forward, from rest, in a straight line along a horizontal floor with acceleration

1.5\ \text{m s}^{-2}

as shown in the diagram below.As it moves the tractor experiences a total resistance force of

1.2

N
(a) Show that the total resistance force experienced by the cart is approximately

1.6

N

(4 marks)
(b) At the instant that the toy reaches a speed of

2.4\ \text{m s}^{-1}

the string breaks.

The tractor and cart then slow down with constant deceleration, travelling a further

h

metres before coming to rest.

You may assume that the resistance forces on the tractor and on the cart do not change.
(b) (i) Find the tension in the rod after the string has broken.

(ii) Find

h

(7 marks)
(c) State one modelling assumption that you have used about the rod when answering part (b)(i).

(1 mark)

●●●●●Level 512 marksStart

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

Two drones,

A

and

B

, are flying at the same constant height. Each drone is moving with constant acceleration and the drones are modelled as particles.

At time

t

seconds, drone

A

has velocity

\mathbf{v}_A\ \text{m s}^{-1}

, where

\mathbf{v}_A = 4\mathbf{i} + 2\mathbf{j} + (-\mathbf{i} + 3\mathbf{j})t

(a) Find the magnitude of the acceleration of

A

.

(2 marks)

When

t = 1

, the velocity of

B

(5\mathbf{i} + 2\mathbf{j})\ \text{m s}^{-1}

When

t = 4

, the velocity of

B

(-\mathbf{i} + 5\mathbf{j})\ \text{m s}^{-1}

(b) Find the acceleration of

B

, giving your answer in terms of

\mathbf{i}

and

\mathbf{j}

.

(2 marks)
(c) Find the velocity of

B

at time

t = 0

, giving your answer in terms of

\mathbf{i}

and

\mathbf{j}

.

(2 marks)

At time

T_1

seconds, where

T_1 > 0

, both drones are moving with the same speed.
(d) Find the exact value of

T_1

(4 marks)

At time

T_2

seconds, where

T_2 > 0

, both drones are moving in the same direction.
(e) Show that

5T_2^2 + pT_2 + q = 0

, where

p

and

q

are integers to be found.

(3 marks)

●●●●●Level 513 marksStart

Vertical motion under gravity

14 questions

LessonNot started

A ball is dropped from rest. Find its speed after

2

s. (

g = 9.8

●●●●●Level 22 marksStart

A stone is dropped from rest. Find the distance it falls in

3

s. (

g = 9.8

●●●●●Level 22 marksStart

A parcel of mass

1.5

kg is released from rest from a drone that is hovering at a height of

40

m above horizontal ground.

In a first model, the parcel is modelled as a particle moving freely under gravity.
(a) Using this first model, find the time taken for the parcel to reach the ground.

(3 marks)

In a second model, the parcel is modelled as a particle falling against a constant resistance to motion of magnitude

R

newtons. Using this second model, the parcel takes

3.5

s to reach the ground.
(b) Find the value of

R

.

(5 marks)

●●●●●Level 38 marksStart

A pebble is projected vertically upwards with speed

U\ \text{m s}^{-1}

from a point

A

on a bridge. The point

A

22.4

m vertically above the surface of a river.

The pebble moves freely under gravity and hits the surface of the river

4

s after it is projected.
(a) Find the value of

U

●●●●●Level 38 marksStart

A small ball of mass

0.2

kg is released from rest at a point

2.5

m above a horizontal floor. The ball falls freely under gravity, hits the floor and rebounds vertically upwards. The coefficient of restitution between the ball and the floor is

0.6

The ball is modelled as a particle.
(a) Find the speed of the ball immediately before it hits the floor.

(2 marks)
(b) Find the magnitude of the impulse exerted on the ball by the floor.

(3 marks)
(c) Find the greatest height above the floor reached by the ball after it rebounds.

(2 marks)

●●●●●Level 37 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

d

metres.

At the same instant both drops begin to fall freely under gravity.

The first drop hits the ground after

1.6

seconds.

The second drop hits the ground

0.3

seconds later.

Show that

d

is approximately

5.1

(4 marks)

●●●●●Level 34 marksStart

A ball is thrown vertically upwards at

14.7\ \text{m s}^{-1}

. Find the time to reach its highest point. (

g = 9.8

●●●●●Level 32 marksStart

A ball is thrown vertically upwards at

19.6\ \text{m s}^{-1}

. Find the greatest height reached. (

g = 9.8

●●●●●Level 32 marksStart

A ball is thrown vertically upwards at

24.5\ \text{m s}^{-1}

from ground level. Find the total time before it returns to the ground. (

g = 9.8

●●●●●Level 32 marksStart

A stone is dropped from a height of

44.1

m. Find the time it takes to reach the ground. (

g = 9.8

●●●●●Level 32 marksStart

A ball is thrown vertically downwards at

5\ \text{m s}^{-1}

from a window. Find
(a) its speed and (b) the distance fallen after

2

s. (

g = 9.8

●●●●●Level 34 marksStart

A small stone is projected vertically upwards with speed

16.8\ \text{m s}^{-1}

from a point

A

at the top of a vertical cliff. The point

A

38.5

m vertically above the point

B

. The point

B

lies on horizontal ground at the foot of the cliff.

The stone moves freely under gravity until it hits the ground at

B

with speed

V\ \text{m s}^{-1}

. After hitting the ground the stone does not rebound.
(a) Find the value of

V

.

(3 marks)
(b) Find the time taken for the stone to reach

B

.

(3 marks)

The point

C

9.8

m vertically above

A

.
(c) Find the length of time for which the stone is above

C

.

(4 marks)
(d) Sketch a speed-time graph for the motion of the stone from the instant it is projected to the instant it reaches

B

.

(No further calculations are required.)

(2 marks)

●●●●●Level 412 marksStart

A small stone is released from rest at the top of a deep well and falls freely towards the water at the bottom of the well.

The stone passes through two different fixed points,

A

and

B

, before reaching the water.

At

A

the stone has velocity

u\ \text{m s}^{-1}

B

the stone has velocity

4u\ \text{m s}^{-1}

It can be assumed that:

* the motion is due to gravitational force only * the acceleration due to gravity remains constant throughout.
(a) Show that the time taken for the stone to travel from

A

B

\dfrac{3u}{g}

seconds.

(2 marks)
(b) Point

A

h

metres above the water.

Show that

h > \frac{15u^2}{2g}

Fully justify your answer.

(3 marks)

●●●●●Level 45 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A machine for cricket catching practice fires cricket balls upwards. Each ball leaves the machine at ground level with a speed of

11.2\ \text{m s}^{-1}

The manufacturer claims that the balls can reach a maximum height of

6.4

0

and

30

degrees from the vertical.

The range of maximum heights,

h

metres, above the ground which can be reached by the balls may be expressed as

k < h \le 6.4

Find the value of

k

(4 marks)

●●●●●Level 48 marksStart

Kinematics with calculus (a → v → s)

34 questions

LessonNot started

A particle moves in a straight line with velocity

v = \left(3t^2 - 4\right)

m s⁻¹ at time

t

seconds. Identify the magnitude of its acceleration when

t = 2

.

Circle your answer.

A $8$ m s⁻²
B $12$ m s⁻²
C $6$ m s⁻²
D $2$ m s⁻²

●●●●●Level 11 markStart

The displacement,

r

metres, of a particle from a fixed point

O

at time

t

seconds is

r = 7.2t - 0.8t^2

for

t \ge 0

(a) Find the value of

r

when

t = 2

(1 mark)
(b) Determine the range of values of

t

for which the displacement is positive.

(2 marks)

●●●●●Level 23 marksStart

A particle moves in a straight line with velocity

v = (4t - t^{2})\ \text{m s}^{-1}

at time

t

seconds.

Find the distance travelled between

t = 0

and

t = 3

. (3 marks)

●●●●●Level 23 marksStart

A particle

P

moves in a straight line. At time

t

seconds (

0 \le t \le 4

), its velocity

v

m s⁻¹ is given by

v = 12t - 3t^2

.
(a) Find the acceleration of

P

when

t = 3

. (2 marks)
(b) Find the displacement of

P

from its starting position when

t = 4

. (3 marks)
(c) Determine the maximum velocity of

P

. (2 marks)

●●●●●Level 37 marksStart

A motorcyclist has an initial velocity of

2\ \text{m s}^{-1}

The motorcyclist is moving in a straight line along a horizontal road.

The acceleration,

a\ \text{m s}^{-2}

, of the motorcyclist at time

t

seconds is given by

a = 1.2 - 4kt + 9kt^2

where

k

is a constant.

When

t = 5

the motorcyclist has a velocity of

21\ \text{m s}^{-1}

Show that

k = \dfrac{1}{25}

(4 marks)

●●●●●Level 34 marksStart

A particle moves in a straight line with acceleration

a = (6t - 4)\ \text{m s}^{-2}

. At

t = 0

its velocity is

2\ \text{m s}^{-1}

.
(a) Find an expression for the velocity at time

t

. (3 marks)
(b) Find the velocity when

t = 2

. (1 mark)

●●●●●Level 34 marksStart

A particle moves in a straight line with velocity

v = (2t + 3)\ \text{m s}^{-1}

. At

t = 0

its displacement from a fixed point

O

5\text{ m}

.
(a) Find an expression for the displacement from

O

at time

t

. (3 marks)
(b) Find the displacement when

t = 4

. (1 mark)

●●●●●Level 34 marksStart

A particle moves in a straight line with velocity

v = (t^{2} - 5t + 6)\ \text{m s}^{-1}

at time

t

seconds.
(a) Find the times at which the particle is instantaneously at rest. (2 marks)
(b) Find the displacement of the particle from its starting point when

t = 4

. (3 marks)
(c) Find the total distance travelled between

t = 0

and

t = 4

. (4 marks)

●●●●●Level 39 marksStart

A particle moves in a straight line with velocity

v = \dfrac{12}{(t + 1)^{2}}\ \text{m s}^{-1}

at time

t

seconds.
(a) Find the acceleration when

t = 1

. (3 marks)
(b) Find the distance travelled between

t = 0

and

t = 1

. (3 marks)

●●●●●Level 36 marksStart

A particle moves in a straight line with velocity

v = (12t - 3t^{2})\ \text{m s}^{-1}

at time

t

seconds.

Find the maximum velocity of the particle. (4 marks)

●●●●●Level 34 marksStart

A particle sets off from rest and moves in a straight line. For

0 \leqslant t \leqslant 2

its velocity is

v = 3t^{2}\ \text{m s}^{-1}

; for

t > 2

it moves at the constant velocity it has reached at

t = 2

.

Find the total distance travelled in the first

5

seconds. (5 marks)

●●●●●Level 35 marksStart

A particle starts from rest at a point

O

and moves in a straight line with acceleration

a = (6 - 2t)\ \text{m s}^{-2}

at time

t

seconds.
(a) Find an expression for the velocity at time

t

, and the maximum velocity. (3 marks)
(b) Find the time at which the particle returns to instantaneous rest. (1 mark)
(c) Find the distance travelled up to that time. (2 marks)

●●●●●Level 36 marksStart

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

of mass

0.5

kg moves on a smooth horizontal plane. For

0 \le t \le 2

, its position vector relative to a fixed origin

O

at time

t

seconds is

\mathbf{r}

metres, where

\mathbf{r} = (t^3 - 8t)\mathbf{i} + (t - t^2)\mathbf{j}

(a) Show that the speed of

P

at the instant when

t = 2

5\text{ m s}^{-1}

(3 marks)

At the instant when

t = 2

P

receives an impulse of

(-8\mathbf{i} - \mathbf{j})

N s.
(b) Find the angle through which the direction of motion of

P

is turned by the impulse.

(4 marks)
(c) Find the increase in the kinetic energy of

P

caused by the impulse.

(2 marks)

●●●●●Level 49 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A single force

\mathbf{F}

N acts on a particle

P

, which moves on a smooth horizontal surface. The velocity of

P

at time

t

seconds, where

t \geq 0

, is

\mathbf{v}\ \text{m s}^{-1}

, where

\mathbf{v} = (t^2 - 4t + 7)\mathbf{i} + (3t - 5)\mathbf{j}.

(a) Show that

P

is never stationary.

(2 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the acceleration of

P

at time

t

.

(1 mark)

The mass of

P

0.6

kg.
(c) Determine the magnitude of

\mathbf{F}

when

P

is moving in the direction of the vector

3\mathbf{i} - 2\mathbf{j}

. Give your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 48 marksStart

A train travels along a straight horizontal track, passing three signals,

A

B

and

C

, in that order. At time

t

seconds after passing

A

, the velocity of the train is

v\text{ m s}^{-1}

, where

v

is modelled by the equation

v = at^2 + bt + c,

where

a

b

and

c

are constants.

The train passes

A

with velocity

12\text{ m s}^{-1}

.
(a) State the value of

c

.

(1 mark)

The train passes

B

when

t = 8

, and at that instant its deceleration is

0.5\text{ m s}^{-2}

. The train passes

C

when

t = 20

, with velocity

4.8\text{ m s}^{-1}

.
(b) Determine the values of

a

and

b

.

(4 marks)
(c) Find, to the nearest metre, the distance between signals

A

and

C

.

(2 marks)

●●●●●Level 47 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A particle

P

moves along a straight line. At time

t = 0

P

is at rest at the fixed point

O

on the line.

At time

t

seconds,

t \geqslant 0

, the acceleration of

P

, in the direction of increasing displacement from

O

, is

k(5 - 2t)

m s

^{-2}

, where

k

is a positive constant.

Given that the speed of

P

18

m s

^{-1}

when

t = 2

(a) show that

k = 3

(3 marks)

At time

t

seconds, the displacement of

P

from

O

s

metres.
(b) Show that

s = \frac{1}{2}t^2(15 - 2t)

(2 marks)
(c) Find the total distance travelled by

P

before it returns to

O

.

(4 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

[In this question,

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

moves on a smooth horizontal plane. At time

t

seconds,

t \geqslant 0

, the position vector of

P

relative to a fixed origin

O

\mathbf{r}

metres, where

\mathbf{r} = (2t^2 - 5t)\mathbf{i} + (t^3 - 6t^2 + 9t)\mathbf{j}

(a) Find the speed of

P

at the instant when

P

is moving in the direction of the vector

\mathbf{i}

(5 marks)
(b) Find the exact magnitude of the acceleration of

P

when

t = 3.5

(4 marks)
(c) Show that

P

never returns to

O

(2 marks)

●●●●●Level 411 marksStart

A remote-controlled car moves on a horizontal surface. The unit vectors

\mathbf{i}

and

\mathbf{j}

are directed east and north respectively.

At time

t

seconds, where

0 \le t \le 5

, the velocity of the car is

\mathbf{v} = \left(t^2 - 4t + 1\right)\mathbf{i} + (2t - 5)\mathbf{j}

m s⁻¹.
(a) Find the acceleration of the car when

t = 2

, and state the direction in which it acts. (2 marks)
(b) Find the value of

t

at which the car is moving in the direction of the vector

\mathbf{i} + \mathbf{j}

. Fully justify your answer. (4 marks)
(c) Find the distance of the car from its starting position when

t = 5

. (3 marks)

●●●●●Level 49 marksStart

A particle

P

of mass

0.5

kg moves in a horizontal plane under the action of a single force

\mathbf{F}

newtons.

At time

t

seconds the position vector of

P

\mathbf{r} = \left(t^3 - 3t\right)\mathbf{i} + 2t^2\,\mathbf{j} \quad \text{metres}

(a) Find the velocity of

P

when

t = 2

. (3 marks)
(b) Find the speed of

P

when

t = 2

, giving your answer to

3

significant figures. (2 marks)
(c) Find the magnitude of

\mathbf{F}

when

t = 2

, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 49 marksStart

A particle

P

moves in a horizontal plane. At time

t

seconds (

t \ge 0

), its position vector relative to a fixed origin

O

\mathbf{r} = (2t^2 - 3t)\mathbf{i} + (4t - t^2)\mathbf{j}

metres.
(a) Find the velocity

\mathbf{v}

P

at time

t

, and show that the acceleration of

P

is constant. (3 marks)
(b) Find the speed of

P

when

t = 3

. (2 marks)
(c) Find the value of

t

at which

P

is moving parallel to

\mathbf{i}

. (2 marks)
(d) Find the distance of

P

from

O

at this time, giving your answer in exact form. (3 marks)

●●●●●Level 410 marksStart

A particle

P

moves along a straight line. At time

t

seconds, where

t \geqslant 0

, the velocity of

P

v

m s⁻¹, where

v = 3t^2 - 12t + 9

t = 0

P

is at the point

O

on the line.
(a) Find the values of

t

at which

P

is instantaneously at rest. (2 marks)
(b) Find the total distance travelled by

P

in the first

4

seconds. (6 marks)
(c) Find the magnitude of the acceleration of

P

when

t = 4

. (2 marks)

●●●●●Level 410 marksStart

A particle

P

moves in a horizontal plane. At time

t

seconds, where

t \geqslant 0

, the acceleration of

P

(6t\,\mathbf{i} + 4\mathbf{j})

m s⁻². When

t = 0

P

is at the point

A

with position vector

(7\mathbf{i} - 5\mathbf{j})

m relative to a fixed origin

O

, and the velocity of

P

(-5\mathbf{i} - \mathbf{j})

m s⁻¹.
(a) Show that

P

is never instantaneously at rest. (3 marks)
(b) Find the value of

t

at which the velocity of

P

is perpendicular to its acceleration, giving a reason for rejecting any other solutions of your equation. (4 marks)
(c) Find the distance of

P

from

O

at this time. (3 marks)

●●●●●Level 410 marksStart

A particle

P

moves along a straight line. At time

t

seconds, where

t \geqslant 0

, the acceleration of

P

a

m s⁻², where

a = 6t - 4

When

t = 0

P

is at the origin

O

and is moving with velocity

5

m s⁻¹.
(a) Find an expression for the velocity of

P

at time

t

. (2 marks)
(b) Find the displacement of

P

from

O

when

t = 3

. (3 marks)
(c) Show that

P

never comes to rest. (2 marks)

●●●●●Level 47 marksStart

A particle is moving in a straight line through the origin

O

The displacement of the particle,

r

metres, from

O

, at time

t

seconds is given by

r = p + 1.5t - q\text{e}^{-0.4t}

where

p

and

q

are constants.

When

t = 3

, the acceleration of the particle is

-0.5\ \text{m s}^{-2}

(a) Show that

q \approx 10.4

(5 marks)
(b) The particle has an initial displacement of

-6

metres.

Find the value of

p

Give your answer to two significant figures.

(2 marks)

●●●●●Level 47 marksStart

A particle moves in a plane with velocity

\mathbf{v} = (6t - 4)\,\mathbf{i} + 3t^2\,\mathbf{j}\ \text{m s}^{-1}

. When

t = 0

its position vector relative to a fixed origin

O

(2\mathbf{i} + \mathbf{j})

m. Find
(a) its position vector at time

t

,
(b) its position vector when

t = 2

, and (c) its distance from

O

when

t = 2

●●●●●Level 45 marksStart

A particle moves in a plane with acceleration

\mathbf{a} = 2t\,\mathbf{i} + (6t^2 - 4)\,\mathbf{j}\ \text{m s}^{-2}

. When

t = 0

its velocity is

(3\mathbf{i} + \mathbf{j})\ \text{m s}^{-1}

. Find
(a) its velocity at time

t

,
(b) its velocity when

t = 2

, and (c) its speed when

t = 2

●●●●●Level 45 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

[In this question, $\mathbf{i}$ is a unit vector due east and $\mathbf{j}$ is a unit vector due north. Position vectors are given relative to a fixed origin $O$ .]

At time

t

seconds,

t \ge 1

, the position vector of a particle

P

\mathbf{r}

metres, where

\mathbf{r} = (2t^3 - 8t^2 + c)\,\mathbf{i} + (t^3 - 2t^2)\,\mathbf{j}

and

c

is a constant.

When

t = 3

, the bearing of

P

from

O

045^\circ

(a) Show that

c = 27

(3 marks)
(b) Find the speed of

P

when

t = 3

(4 marks)

When

t = T

P

is accelerating in the direction of

(2\mathbf{i} + 5\mathbf{j})

.
(c) Find the value of

T

.

(4 marks)

●●●●●Level 511 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

1.2

kg is moving on a smooth horizontal surface under the action of a single horizontal force

\mathbf{F}

N. At time

t

seconds, where

t \geq 0

, the velocity

\mathbf{v}\ \text{m s}^{-1}

P

, relative to a fixed origin

O

, is given by

\mathbf{v} = (t^2 - 8t + 20)\mathbf{i} + (t - 6)\mathbf{j}.

(a) Show that

P

is never stationary.

(2 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the acceleration of

P

at time

t

.

(1 mark)
(c) Determine the magnitude of

\mathbf{F}

when

P

is moving in the direction of the vector

5\mathbf{i} - \mathbf{j}

. Give your answer correct to 3 significant figures.

(5 marks)

When

t = 3

P

is at the point with position vector

(-9\mathbf{i} + 1.5\mathbf{j})

m.
(d) Determine the bearing of

P

from

O

at time

t = 6

.

(5 marks)

●●●●●Level 513 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

2.5

kg moves on a smooth horizontal surface. At time

t

seconds, where

t \geq 0

, the only horizontal forces acting on

P

are a constant force

(-9\mathbf{i} - 12.5\mathbf{j})

N and a force

\left(3t\mathbf{i} + (1.5t^2 - 1)\mathbf{j}\right)

N which varies with time.
(a) Determine the value of

t

when the forces acting on

P

are in equilibrium.

(2 marks)

It is given that

P

is at rest when

t = 0

.
(b) Determine the speed of

P

at the instant when

P

is moving due north.

(6 marks)
(c) Determine the distance between the positions of

P

when

t = 0

and

t = 5

.

(5 marks)

●●●●●Level 513 marksStart

In this question you should take the acceleration due to gravity to be

10

m s

^{-2}

.A small ball

P

is thrown from a point

A

on a balcony with speed

15

m s

^{-1}

at an angle of elevation

\theta

, where

\sin\theta = \frac{4}{5}

and

\cos\theta = \frac{3}{5}

. The point

A

9

m vertically above a point

B

on horizontal ground. The ball first lands at a point

C

on the ground (see diagram).

P

is modelled as a particle moving freely under gravity.
(a) Find the maximum height of

P

above the ground during its motion.

(3 marks)

The time taken for

P

to travel from

A

C

T

seconds.
(b) Determine the value of

T

.

(3 marks)
(c) State one limitation of the model, other than air resistance, that could affect the answer to part (b).

(1 mark)

At the instant that

P

is thrown, a small toy car

Q

starts from rest at

B

and moves in a straight line along the ground towards

C

. At time

t

seconds, where

t \ge 0

, the velocity

v

m s

^{-1}

Q

is given by

v = kt^2 + 1.2t

, where

k

is a positive constant.
(d) Given that

P

and

Q

collide at

C

, determine the acceleration of

Q

immediately before this collision.

(6 marks)

●●●●●Level 513 marksStart

A van starts from rest and moves along a straight horizontal road. For the first 15 seconds of the motion, the velocity of the van,

v\text{ m s}^{-1}

, at time

t

seconds after it starts is given by

v = kt - 0.02t^2,

where

k

is a constant.

At the instant when

t = 15

the acceleration of the van is

0.9\text{ m s}^{-2}

. From

t = 15

the van moves with constant acceleration

0.9\text{ m s}^{-2}

until its speed is

22\text{ m s}^{-1}

.
(a) Find the value of

k

.

(3 marks)
(b) Find the total distance travelled by the van from the start until its speed is

22\text{ m s}^{-1}

.

(7 marks)

●●●●●Level 510 marksStart

A particle

P

moves in a straight line. The velocity

v\text{ m s}^{-1}

P

at time

t

seconds is given by

v = \tfrac{1}{16}kt(t - 4) \quad \text{for } 0 \leqslant t \leqslant 6,

v = \frac{27k}{t^2} \quad \text{for } 6 \leqslant t \leqslant 10,

where

k

is a positive constant.
(a) Describe the velocity–time graph for

P

for values of

t

from 0 to 10. You should state the values of

t

at which the graph meets the

t

-axis and the values of

v

, in terms of

k

, when

t = 6

and when

t = 10

, and describe the shape of each section of the graph.

(3 marks)
(b) Find the value of

t

, where

0 \leqslant t \leqslant 10

, at which

P

has zero acceleration.

(1 mark)
(c) In this question you must show detailed reasoning.

It is given that

P

travels a total distance of

94

m between

t = 0

and

t = 10

.

Find the value of

k

.

(6 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A particle

P

moves along a straight line. The speed of

P

at time

t

seconds is

v

m s

^{-1}

, where

v

is the continuous function

v = \begin{cases} t^2 + 3t + 2 & 0 \leqslant t \leqslant k \\ 6t + 12 & k < t \leqslant 10 \end{cases}

and

k

is a constant.
(a) Show that

k = 5

, explaining your method carefully.

(3 marks)
(b) Find the acceleration of

P

when

t = 2.5

(3 marks)

When

t = 0

P

passes through the fixed point

O

on the line.
(c) Find the distance of

P

from

O

when

t = 8

(7 marks)

●●●●●Level 513 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

of mass

0.5

kg moves on a smooth horizontal plane. A single horizontal force acts on

P

, so that at time

t

seconds, for

0 \le t \le 2

, the velocity of

P

\mathbf{v}\text{ m s}^{-1}

, where

\mathbf{v} = (3t^2 - 1)\mathbf{i} + 4t\,\mathbf{j}

At time

t = 0

P

is at the fixed origin

O

.
(a) Show that the position vector of

P

relative to

O

when

t = 2

(6\mathbf{i} + 8\mathbf{j})

m.

(3 marks)

When

t = 2

, the force stops acting and

P

receives an impulse. Immediately after receiving the impulse,

P

moves with constant velocity directly towards

O

, with speed

5\text{ m s}^{-1}

.
(b) Find the magnitude of the impulse.

(6 marks)
(c) Find the kinetic energy lost by

P

as a result of the impulse.

(3 marks)
(d) Find the value of

t

when

P

reaches

O

.

(2 marks)

●●●●●Level 514 marksStart

Motion in two dimensions with vectors

48 questions

LessonNot started

A particle

P

of mass

2

kg moves on a smooth horizontal surface with velocity

(4\mathbf{i} + 3\mathbf{j})\text{ m s}^{-1}

, where

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors. It receives an impulse of

(-12\mathbf{i} + 4\mathbf{j})

N s.

Find the angle through which the direction of motion of

P

is turned by the impulse.

(4 marks)

●●●●●Level 24 marksStart

Two particles,

P

and

Q

, are moving in separate straight lines across a smooth horizontal surface.

P

moves with constant velocity

(4\mathbf{i} - 6\mathbf{j})\ \text{m s}^{-1}

Q

moves from the point with position vector

(-3\mathbf{i} + 5\mathbf{j})

metres to the point with position vector

(7\mathbf{i} - 10\mathbf{j})

metres.

Show that

P

and

Q

move along parallel lines.

(3 marks)

●●●●●Level 23 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

of mass

0.8

kg is moving on a smooth horizontal surface with velocity

(3\mathbf{i} + 2\mathbf{j})\text{ m s}^{-1}

when it receives an impulse

\lambda(2\mathbf{i} - \mathbf{j})

N s, where

\lambda

is a constant. Immediately after receiving the impulse,

P

is moving parallel to

\mathbf{i}

.
(a) Find the value of

\lambda

.

(4 marks)
(b) Find the speed of

P

immediately after receiving the impulse.

(2 marks)

●●●●●Level 36 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A puck of mass

0.05

kg glides across the smooth horizontal surface of an air-hockey table with velocity

(6\mathbf{i} + 2\mathbf{j})\text{ m s}^{-1}

. A player hits the puck, and it then moves with velocity

(-6\mathbf{i} + 11\mathbf{j})\text{ m s}^{-1}

. The puck is modelled as a particle.
(a) Find the magnitude of the impulse that the player gives the puck.

(4 marks)
(b) Find the angle between the direction of this impulse and the direction in which the puck was moving before it was hit.

(3 marks)

●●●●●Level 37 marksStart

A particle

P

moves with constant acceleration

(1.5\mathbf{i} - 4\mathbf{j})\ \text{m s}^{-2}

. At time

t = 0

seconds,

P

is moving with velocity

(-2\mathbf{i} + 7\mathbf{j})\ \text{m s}^{-1}

.
(a) Determine the speed of

P

when

t = 4

.

(4 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the displacement of

P

from its position at

t = 0

to its position at

t = 4

.

(2 marks)

●●●●●Level 36 marksStart

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$ .]

Two remote-controlled model yachts,

P

and

Q

, are sailing on a large lake. Each yacht is moving with constant velocity and the yachts are modelled as particles.

At time

t

seconds, the position vector of

P

\mathbf{r}

metres.

When $t = 4$ , $\mathbf{r} = (-3\mathbf{i} + 14\mathbf{j})$
When $t = 10$ , $\mathbf{r} = (15\mathbf{i} + 2\mathbf{j})$

(a) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the velocity of

P

.

(2 marks)
(b) Find an expression for

\mathbf{r}

in terms of

t

\mathbf{i}

and

\mathbf{j}

.

(2 marks)

At time

t

seconds, the position vector of

Q

\mathbf{s}

metres, where

\mathbf{s} = 8\mathbf{i} - 5\mathbf{j} + t(-\mathbf{i} + 4\mathbf{j})

(c) Find, to the nearest degree, the bearing of

Q

from

P

when

t = 10

(3 marks)

●●●●●Level 37 marksStart

A particle moves in a plane so that at time

t

seconds its position vector is

\mathbf{r} = (t^3 - 2t)\,\mathbf{i} + (3t^2 - t)\,\mathbf{j}

metres. Find
(a) its velocity at time

t

,
(b) its acceleration at time

t

, and (c) its speed when

t = 1

●●●●●Level 35 marksStart

A particle moves with constant acceleration

\mathbf{a} = (-\mathbf{i} + 2\mathbf{j})\ \text{m s}^{-2}

. At

t = 0

its velocity is

\mathbf{u} = (4\mathbf{i} - 3\mathbf{j})\ \text{m s}^{-1}

. Find
(a) its velocity when

t = 3

,
(b) its speed when

t = 3

, and (c) the value of

t

at which it is moving parallel to

\mathbf{j}

●●●●●Level 34 marksStart

A particle moves in a plane so that at time

t

seconds its position vector is

\mathbf{r} = (3t^{2} - 1)\,\mathbf{i} + (2t + 4)\,\mathbf{j} \text{ metres}

Find the speed of the particle when

t = 2

, giving your answer in the form

k\sqrt{37}

m s

^{-1}

. (4 marks)

●●●●●Level 34 marksStart

A particle has initial velocity

(2\mathbf{i} - 3\mathbf{j})

m s

^{-1}

and moves with constant acceleration

(4\mathbf{i} + \mathbf{j})

m s

^{-2}

.

Find its velocity after

3

seconds, and describe the direction of motion at that instant. (3 marks)

●●●●●Level 33 marksStart

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$ .]

A particle

P

is moving on a smooth horizontal plane.

The particle has constant acceleration

(2\mathbf{i} + 3\mathbf{j})\ \text{m s}^{-2}

At time

t = 0

P

passes through the point

A

.

At time

t = 3

P

passes through the point

B

.

The velocity of

P

as it passes through

A

(5\mathbf{i} - 11\mathbf{j})\ \text{m s}^{-1}

(a) Find the speed of

P

as it passes through

B

.

(4 marks)

The position vector of

A

(-9\mathbf{i} - 20\mathbf{j})

m.

At time

t = T

seconds, where

T > 3

P

passes through the point

C

.

The position vector of

C

(41\mathbf{i} + c\mathbf{j})

m.
(b) Find the value of

T

.

(3 marks)
(c) Find the value of

c

.

(3 marks)

●●●●●Level 410 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

of mass

2

kg moves on a smooth horizontal surface with velocity

5\mathbf{j}\text{ m s}^{-1}

. It receives an impulse

\lambda(\mathbf{i} - \mathbf{j})

N s, where

\lambda

is a positive constant, and as a result it loses

10.5

J of kinetic energy.
(a) Show that

\lambda^2 - 10\lambda + 21 = 0

(5 marks)

Given also that the magnitude of the impulse is less than

6

N s,
(b) find the velocity of

P

immediately after the impulse, giving your answer in terms of

\mathbf{i}

and

\mathbf{j}

.

(3 marks)
(c) Find the angle through which the impulse turns the direction of motion of

P

.

(2 marks)

●●●●●Level 410 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A box of mass

2.5

kg slides across a smooth horizontal floor with velocity

(2\mathbf{i} - 3\mathbf{j})\text{ m s}^{-1}

. Someone kicks the box. The kick gives it a horizontal impulse

\mathbf{I}

N s, where

|\mathbf{I}| = 10\sqrt2

, and the box then slides with velocity

(\lambda\mathbf{i} + \mathbf{j})\text{ m s}^{-1}

, where

\lambda

is a positive constant. The box is modelled as a particle.
(a) Show that

(\lambda - 2)^2 = 16

(4 marks)
(b) Hence find

\mathbf{I}

in terms of

\mathbf{i}

and

\mathbf{j}

.

(3 marks)
(c) Find the angle between

\mathbf{I}

and the direction in which the box was sliding before the kick.

(2 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

of mass

0.5

kg moves on a smooth horizontal plane. For

0 \le t \le 2

, its position vector relative to a fixed origin

O

at time

t

seconds is

\mathbf{r}

metres, where

\mathbf{r} = (t^3 - 8t)\mathbf{i} + (t - t^2)\mathbf{j}

(a) Show that the speed of

P

at the instant when

t = 2

5\text{ m s}^{-1}

(3 marks)

At the instant when

t = 2

P

receives an impulse of

(-8\mathbf{i} - \mathbf{j})

N s.
(b) Find the angle through which the direction of motion of

P

is turned by the impulse.

(4 marks)
(c) Find the increase in the kinetic energy of

P

caused by the impulse.

(2 marks)

●●●●●Level 49 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

of mass

0.6

kg is moving on a smooth horizontal surface with velocity

(2\mathbf{i} + \mathbf{j})\text{ m s}^{-1}

when it receives an impulse

\lambda(-\mathbf{i} + \mathbf{j})

N s, where

\lambda

is a constant. Immediately after receiving the impulse,

P

is moving in a direction perpendicular to its direction of motion immediately before it received the impulse.
(a) Show that

\lambda = 3

(5 marks)
(b) Find the increase in the kinetic energy of

P

caused by the impulse.

(3 marks)

●●●●●Level 48 marksStart

A particle

P

of mass

m

kg is moving on a smooth horizontal surface under the action of two constant horizontal forces

(3\mathbf{i} - 5\mathbf{j})

N and

(a\mathbf{i} + b\mathbf{j})

N. The resultant of these two forces is

\mathbf{R}

N. It is given that

\mathbf{R}

acts in a direction which is parallel to the vector

2\mathbf{i} + \mathbf{j}

.
(a) Show that

a - 2b + 13 = 0

.

(3 marks)

It is given that

a = 5

and that

P

moves with an acceleration of magnitude

1.25\ \text{m s}^{-2}

.
(b) Determine the value of

m

.

(4 marks)

●●●●●Level 47 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A single force

\mathbf{F}

N acts on a particle

P

, which moves on a smooth horizontal surface. The velocity of

P

at time

t

seconds, where

t \geq 0

, is

\mathbf{v}\ \text{m s}^{-1}

, where

\mathbf{v} = (t^2 - 4t + 7)\mathbf{i} + (3t - 5)\mathbf{j}.

(a) Show that

P

is never stationary.

(2 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the acceleration of

P

at time

t

.

(1 mark)

The mass of

P

0.6

kg.
(c) Determine the magnitude of

\mathbf{F}

when

P

is moving in the direction of the vector

3\mathbf{i} - 2\mathbf{j}

. Give your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 48 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

[In this question,

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

moves on a smooth horizontal plane. At time

t

seconds,

t \geqslant 0

, the position vector of

P

relative to a fixed origin

O

\mathbf{r}

metres, where

\mathbf{r} = (2t^2 - 5t)\mathbf{i} + (t^3 - 6t^2 + 9t)\mathbf{j}

(a) Find the speed of

P

at the instant when

P

is moving in the direction of the vector

\mathbf{i}

(5 marks)
(b) Find the exact magnitude of the acceleration of

P

when

t = 3.5

(4 marks)
(c) Show that

P

never returns to

O

(2 marks)

●●●●●Level 411 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are unit vectors, with

\mathbf{i}

horizontal and

\mathbf{j}

vertical.]Figure 1 (not accurately drawn)

Two points,

A

and

B

, are on horizontal ground. A particle

P

is projected from

A

at time

t = 0

with velocity

(6\mathbf{i} + 14\mathbf{j})

m s

^{-1}

and moves freely under gravity until it hits the ground at

B

, as shown in Figure 1.
(a) Find the distance

AB

.

(4 marks)

For an interval of

T

seconds during its flight, the speed of

P

is less than

7.5

m s

^{-1}

.
(b) Find the value of

T

.

(3 marks)

When

P

is moving perpendicular to the direction in which it was projected, its height above the ground is

h

metres.
(c) Find the value of

h

.

(4 marks)

●●●●●Level 411 marksStart

A remote-controlled car moves on a horizontal surface. The unit vectors

\mathbf{i}

and

\mathbf{j}

are directed east and north respectively.

At time

t

seconds, where

0 \le t \le 5

, the velocity of the car is

\mathbf{v} = \left(t^2 - 4t + 1\right)\mathbf{i} + (2t - 5)\mathbf{j}

m s⁻¹.
(a) Find the acceleration of the car when

t = 2

, and state the direction in which it acts. (2 marks)
(b) Find the value of

t

at which the car is moving in the direction of the vector

\mathbf{i} + \mathbf{j}

. Fully justify your answer. (4 marks)
(c) Find the distance of the car from its starting position when

t = 5

. (3 marks)

●●●●●Level 49 marksStart

A particle

P

of mass

0.5

kg moves in a horizontal plane under the action of a single force

\mathbf{F}

newtons.

At time

t

seconds the position vector of

P

\mathbf{r} = \left(t^3 - 3t\right)\mathbf{i} + 2t^2\,\mathbf{j} \quad \text{metres}

(a) Find the velocity of

P

when

t = 2

. (3 marks)
(b) Find the speed of

P

when

t = 2

, giving your answer to

3

significant figures. (2 marks)
(c) Find the magnitude of

\mathbf{F}

when

t = 2

, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 49 marksStart

A particle

P

moves in a horizontal plane. At time

t

seconds (

t \ge 0

), its position vector relative to a fixed origin

O

\mathbf{r} = (2t^2 - 3t)\mathbf{i} + (4t - t^2)\mathbf{j}

metres.
(a) Find the velocity

\mathbf{v}

P

at time

t

, and show that the acceleration of

P

is constant. (3 marks)
(b) Find the speed of

P

when

t = 3

. (2 marks)
(c) Find the value of

t

at which

P

is moving parallel to

\mathbf{i}

. (2 marks)
(d) Find the distance of

P

from

O

at this time, giving your answer in exact form. (3 marks)

●●●●●Level 410 marksStart

A particle

P

moves in a horizontal plane. At time

t

seconds, where

t \geqslant 0

, the acceleration of

P

(6t\,\mathbf{i} + 4\mathbf{j})

m s⁻². When

t = 0

P

is at the point

A

with position vector

(7\mathbf{i} - 5\mathbf{j})

m relative to a fixed origin

O

, and the velocity of

P

(-5\mathbf{i} - \mathbf{j})

m s⁻¹.
(a) Show that

P

is never instantaneously at rest. (3 marks)
(b) Find the value of

t

at which the velocity of

P

is perpendicular to its acceleration, giving a reason for rejecting any other solutions of your equation. (4 marks)
(c) Find the distance of

P

from

O

at this time. (3 marks)

●●●●●Level 410 marksStart

Two particles,

P

and

Q

, are moving in separate straight lines across a smooth horizontal surface.

P

moves with constant velocity

(8\mathbf{i} - 6\mathbf{j})\ \text{m s}^{-1}

Q

moves from the point with position vector

(-5\mathbf{i} + 9\mathbf{j})

metres to the point with position vector

(15\mathbf{i} - 6\mathbf{j})

metres during a

2.5

second period.
(a) Show that

P

and

Q

move along parallel lines.

(3 marks)
(b) Megan says

“

Q

is also moving with a constant velocity of

(8\mathbf{i} - 6\mathbf{j})\ \text{m s}^{-1}

”

Explain why Megan may be incorrect.

(1 mark)
(c) A third particle,

R

, is moving with a constant speed of

6\ \text{m s}^{-1}

, in a straight line, across the same surface.

P

and

R

move along lines that intersect at a fixed point

X

.

It is given that:

$P$ passes through $X$ exactly $1$ second after $R$ passes through $X$
$P$ and $R$ are exactly $50$ metres apart $5$ seconds after $R$ passes through $X$

Show that

P

and

R

move along perpendicular lines.

(5 marks)

●●●●●Level 49 marksStart

Two particles,

P

and

Q

, move in the same horizontal plane.

Particle

P

is initially at rest at the point with position vector

(2\mathbf{i} + \mathbf{j})

metres and moves with constant acceleration

(3\mathbf{i} - 2\mathbf{j})\ \text{m s}^{-2}

Particle

Q

moves in a straight line, passing through the points with position vectors

(10\mathbf{i} - 3\mathbf{j})

metres and

(\mathbf{i} + c\mathbf{j})

metres, where

c

is a constant.

P

and

Q

are moving along parallel paths.
(a) Show that

c = 3

(4 marks)
(b) (i) Find an expression for the position vector of

P

at time

t

seconds.

(ii) Hence, prove that the paths of

P

and

Q

are not collinear.

(4 marks)

●●●●●Level 48 marksStart

A particle,

T

, moves around the edges of a triangle

ABC

, in an anticlockwise direction, at a constant speed, as shown in the diagram below.When

T

moves from

A

B

its velocity is

\left(2\sqrt{3}\,\mathbf{i} + 2\mathbf{j}\right)\ \text{m s}^{-1}

(a) Find the speed of

T

.

(1 mark)
(b) When

T

moves from

B

C

its velocity is

\left(-2\sqrt{3}\,\mathbf{i} + 2\mathbf{j}\right)\ \text{m s}^{-1}

Show that the acute angle

ABC = 60^\circ

(2 marks)
(c) It is given that

ABC

is an equilateral triangle.

T

returns to its initial position after

18

seconds.

Vertex

B

lies at the point with position vector

\begin{bmatrix} 5 \\ 1 \end{bmatrix}

metres.

Find the position vector of

C

.

(3 marks)

●●●●●Level 46 marksStart

A particle is moving such that its position vector,

\mathbf{r}

metres, at time

t

seconds, is given by

\mathbf{r} = \mathrm{e}^{2t}\cos t\,\mathbf{i} + \mathrm{e}^{2t}\sin t\,\mathbf{j}

Show that the magnitude of the acceleration of the particle,

a\ \text{m s}^{-2}

, is given by

a = 5\mathrm{e}^{2t}

Fully justify your answer.

(7 marks)

●●●●●Level 47 marksStart

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

A box of mass

1.5

kg is at rest on a smooth horizontal floor. The box is modelled as a particle.

At time

t = 0

, two horizontal forces,

\mathbf{F}_1 = (5\mathbf{i} - 4\mathbf{j})

N and

\mathbf{F}_2 = (p\mathbf{i} + q\mathbf{j})

N, where

p

and

q

are constants, are applied to the box. No other horizontal forces act on the box.

Given that the box moves in the direction of the vector

(3\mathbf{i} + \mathbf{j})

,
(a) show that

p - 3q + 17 = 0

(4 marks)

Given that

q = 7

(b) find the speed of the box at time

t = 3

seconds.

(5 marks)

●●●●●Level 49 marksStart

(80\mathbf{i} + 150\mathbf{j})\ \text{km h}^{-1}

At 14:00, the aircraft is at the point with position vector

(-20\mathbf{i} - 30\mathbf{j})

km.

The aircraft is modelled as a particle moving in a horizontal plane.
(a) Find the speed of the aircraft.

(2 marks)

At time

t

hours after 14:00, the position vector of the aircraft is

\mathbf{a}

km.
(b) Find an expression for

\mathbf{a}

in terms of

\mathbf{i}

\mathbf{j}

and

t

.

(2 marks)

A tall radio mast is modelled as being at the point with position vector

(10\mathbf{i} + 19\mathbf{j})

km in the same horizontal plane.

Aircraft are not permitted to fly within

3

km of the mast.
(c) (i) Find the value of

t

when the aircraft is closest to the mast.

(ii) Hence determine whether the aircraft can continue on its course without breaking this rule.

(7 marks)

●●●●●Level 411 marksStart

A

and

B

, are crossing a large frozen lake. The snowmobiles are modelled as particles.

Snowmobile

A

is moving with constant velocity.

At 10:00, the position vector of

A

(-6\mathbf{i} + 11\mathbf{j})

km.

At 10:40, the position vector of

A

(10\mathbf{i} - \mathbf{j})

km.

At time

t

hours after 10:00, the position vector of

A

\mathbf{r}_A

km.
(a) Show that

\mathbf{r}_A = (-6 + 24t)\mathbf{i} + (11 - 18t)\mathbf{j}

(4 marks)

The speed of

A

V\ \text{m s}^{-1}

(b) Find the value of

V

.

(2 marks)

Snowmobile

B

is moving with constant velocity

(16\mathbf{i} - 20\mathbf{j})\ \text{km h}^{-1}

At 10:00, the position vector of

B

(10\mathbf{i} + 9\mathbf{j})

km.

At 11:15,

B

passes through the point

Q

.
(c) Show that

A

also passes through

Q

.

(5 marks)

●●●●●Level 411 marksStart

At time

t

seconds, where

t \ge 0

, a particle

P

has velocity

\mathbf{v}\ \text{m s}^{-1}

where

\mathbf{v} = (2t^2 - 5t + 1)\,\mathbf{i} + (t^2 - 3t + 9)\,\mathbf{j}.

Find
(a) the speed of

P

at time

t = 0

(3 marks)
(b) the value of

t

when

P

is moving parallel to

(\mathbf{i} + \mathbf{j})

(2 marks)
(c) the acceleration of

P

at time

t

seconds

(2 marks)
(d) the value of

t

when the direction of the acceleration of

P

is perpendicular to

\mathbf{i}

(2 marks)

●●●●●Level 49 marksStart

A particle moves in a plane with velocity

\mathbf{v} = (6t - 4)\,\mathbf{i} + 3t^2\,\mathbf{j}\ \text{m s}^{-1}

. When

t = 0

its position vector relative to a fixed origin

O

(2\mathbf{i} + \mathbf{j})

m. Find
(a) its position vector at time

t

,
(b) its position vector when

t = 2

, and (c) its distance from

O

when

t = 2

●●●●●Level 45 marksStart

A particle moves in a plane with acceleration

\mathbf{a} = 2t\,\mathbf{i} + (6t^2 - 4)\,\mathbf{j}\ \text{m s}^{-2}

. When

t = 0

its velocity is

(3\mathbf{i} + \mathbf{j})\ \text{m s}^{-1}

. Find
(a) its velocity at time

t

,
(b) its velocity when

t = 2

, and (c) its speed when

t = 2

●●●●●Level 45 marksStart

A particle moves in a plane with velocity

\mathbf{v} = (t^2 - 2t)\,\mathbf{i} + (3t - 4)\,\mathbf{j}\ \text{m s}^{-1}

, for

t \ge 0

. Find
(a) the value of

t

at which it is moving parallel to

\mathbf{i}

,
(b) the value of

t

at which it is moving parallel to

\mathbf{j}

, and (c) explain why

t = 0

is not a valid answer to part (b).

●●●●●Level 44 marksStart

At time

t

seconds a particle has velocity

\mathbf{v} = (t^2 - 4t + 3)\,\mathbf{i} + (2t - 4)\,\mathbf{j}\ \text{m s}^{-1}

, for

t \ge 0

. Find
(a) the speed of the particle when

t = 0

,
(b) the values of

t

at which it is moving parallel to

\mathbf{j}

,
(c) the acceleration at time

t

, and (d) the value of

t

at which the acceleration is perpendicular to

\mathbf{i}

●●●●●Level 46 marksStart

A particle starts from the origin with velocity

(3\mathbf{i} + \mathbf{j})

m s

^{-1}

and moves with constant acceleration

(2\mathbf{i} - 4\mathbf{j})

m s

^{-2}

.

Find the distance of the particle from the origin after

2

seconds, giving your answer in the form

k\sqrt{34}

m. (4 marks)

●●●●●Level 44 marksStart

A particle moves with velocity

\mathbf{v} = (2t - 6)\,\mathbf{i} + (4t - 3)\,\mathbf{j}

m s

^{-1}

at time

t

seconds.

Find the time at which the particle is moving parallel to

\mathbf{j}

, and its speed at that instant. (4 marks)

●●●●●Level 44 marksStart

A particle has acceleration

\mathbf{a} = 6t\,\mathbf{i} - 4\,\mathbf{j}

m s

^{-2}

at time

t

seconds, and initial velocity

(\mathbf{i} + 2\mathbf{j})

m s

^{-1}

.

Find the speed of the particle when

t = 1

, giving your answer in the form

k\sqrt{5}

m s

^{-1}

. (5 marks)

●●●●●Level 45 marksStart

A particle has velocity

\mathbf{v} = (3t^{2} - 4)\,\mathbf{i} + 2t\,\mathbf{j}

m s

^{-1}

at time

t

seconds, and is initially at the point with position vector

2\mathbf{i}

m.

Find the position vector of the particle when

t = 2

. (5 marks)

●●●●●Level 45 marksStart

t

seconds,

t \ge 1

, the position vector of a particle

P

\mathbf{r}

metres, where

\mathbf{r} = (2t^3 - 8t^2 + c)\,\mathbf{i} + (t^3 - 2t^2)\,\mathbf{j}

and

c

is a constant.

When

t = 3

, the bearing of

P

from

O

045^\circ

(a) Show that

c = 27

(3 marks)
(b) Find the speed of

P

when

t = 3

(4 marks)

When

t = T

P

is accelerating in the direction of

(2\mathbf{i} + 5\mathbf{j})

.
(c) Find the value of

T

.

(4 marks)

●●●●●Level 511 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

1.2

kg is moving on a smooth horizontal surface under the action of a single horizontal force

\mathbf{F}

N. At time

t

seconds, where

t \geq 0

, the velocity

\mathbf{v}\ \text{m s}^{-1}

P

, relative to a fixed origin

O

, is given by

\mathbf{v} = (t^2 - 8t + 20)\mathbf{i} + (t - 6)\mathbf{j}.

(a) Show that

P

is never stationary.

(2 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the acceleration of

P

at time

t

.

(1 mark)
(c) Determine the magnitude of

\mathbf{F}

when

P

is moving in the direction of the vector

5\mathbf{i} - \mathbf{j}

. Give your answer correct to 3 significant figures.

(5 marks)

When

t = 3

P

is at the point with position vector

(-9\mathbf{i} + 1.5\mathbf{j})

m.
(d) Determine the bearing of

P

from

O

at time

t = 6

.

(5 marks)

●●●●●Level 513 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

2.5

kg moves on a smooth horizontal surface. At time

t

seconds, where

t \geq 0

, the only horizontal forces acting on

P

are a constant force

(-9\mathbf{i} - 12.5\mathbf{j})

N and a force

\left(3t\mathbf{i} + (1.5t^2 - 1)\mathbf{j}\right)

N which varies with time.
(a) Determine the value of

t

when the forces acting on

P

are in equilibrium.

(2 marks)

It is given that

P

is at rest when

t = 0

.
(b) Determine the speed of

P

at the instant when

P

is moving due north.

(6 marks)
(c) Determine the distance between the positions of

P

when

t = 0

and

t = 5

.

(5 marks)

●●●●●Level 513 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

of mass

0.5

kg moves on a smooth horizontal plane. A single horizontal force acts on

P

, so that at time

t

seconds, for

0 \le t \le 2

, the velocity of

P

\mathbf{v}\text{ m s}^{-1}

, where

\mathbf{v} = (3t^2 - 1)\mathbf{i} + 4t\,\mathbf{j}

At time

t = 0

P

is at the fixed origin

O

.
(a) Show that the position vector of

P

relative to

O

when

t = 2

(6\mathbf{i} + 8\mathbf{j})

m.

(3 marks)

When

t = 2

, the force stops acting and

P

receives an impulse. Immediately after receiving the impulse,

P

moves with constant velocity directly towards

O

, with speed

5\text{ m s}^{-1}

.
(b) Find the magnitude of the impulse.

(6 marks)
(c) Find the kinetic energy lost by

P

as a result of the impulse.

(3 marks)
(d) Find the value of

t

when

P

reaches

O

.

(2 marks)

●●●●●Level 514 marksStart

[In this question

\mathbf{i}

and

\mathbf{j}

are perpendicular unit vectors, with

\mathbf{i}

horizontal and

\mathbf{j}

vertically upwards.]

A ball of mass

0.16

kg is moving with velocity

(-20\mathbf{i} - 5\mathbf{j})\text{ m s}^{-1}

when it is struck by a bat at the point

A

, which is

2

m vertically above horizontal ground. The bat gives the ball an impulse of

(6.4\mathbf{i} + 3.2\mathbf{j})

N s. The ball is modelled as a particle.
(a) Show that the velocity of the ball immediately after it is struck is

(20\mathbf{i} + 15\mathbf{j})\text{ m s}^{-1}

(3 marks)
(b) Find the angle through which the direction of motion of the ball is turned by the impulse.

(3 marks)

From

A

the ball moves freely under gravity and first reaches the ground at the point

B

.
(c) Find the horizontal distance from

A

B

.

(4 marks)
(d) Using conservation of mechanical energy, find the speed with which the ball hits the ground.

(3 marks)

●●●●●Level 513 marksStart

A

and a lifeboat

B

are each moving with constant velocity.

the velocity of $A$ is $36\ \text{km h}^{-1}$ due east
the velocity of $B$ is $29\ \text{km h}^{-1}$ on a bearing of $\alpha$ , where $\tan\alpha = \dfrac{21}{20}$