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133 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Motion graphs (displacement–time & velocity–time)

20 questions

LessonNot started

P

moves in a straight line. The velocity

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

P

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,

k

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

P

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

k

t = 6

t = 10

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

(3 marks)
(b) Find the value of

t

0 \leqslant t \leqslant 10

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

t = 0

t = 10

.

Find the value of

k

●●●●●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

t

v

^{-1}

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}

k

is a constant.
(a) Show that

k = 5

, explaining your method carefully.

(3 marks)
(b) Find the acceleration of

P

t = 2.5

(3 marks)

When

t = 0

P

passes through the fixed point

O

on the line.
(c) Find the distance of

P

O

t = 8

●●●●●Level 513 marksStart

A

B

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

A go-kart

P

starts from rest at

A

t = 0

and moves along the track towards

B

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

P

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}

t = 30

P

B

.

Given that the distance

AB

300

m,
(a) show that

V = 12

(3 marks)
(b) Find the acceleration of

P

t = 0

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}

t = 8

, a second go-kart

Q

starts from rest at

A

and moves along the track in the same direction as

P

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}

Q

catches up with

P

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

●●●●●Level 513 marksStart

Constant acceleration (suvat)

54 questions

LessonNot started

A

B

3

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

P

B

at the instant that it hits the ground is

2.5

^{-1}

.
(a) For the motion before

B

hits the ground, show that the acceleration of

B

2.5

^{-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}

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

. Both faces are rough.

A particle

A

6

kg on the first face and a particle

B

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

\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

1.5

^{-1}

the string breaks. In the

0.4

seconds after the string breaks

A

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

A

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}

AB = 2.55

m (see diagram).A particle

P

is projected up the plane with speed

7

^{-1}

A

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

^{-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

A

C

T

seconds.
(b) Determine the value of

T

●●●●●Level 512 marksStart

Ox

Oy

are fixed, with

Ox

horizontal along level ground and

Oy

vertically upwards. The units of

x

y

are metres. A small stone

S

O

with a velocity whose components along

Ox

Oy

U

^{-1}

V

^{-1}

S

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

S

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

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

^{-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

Oy

U

^{-1}

V

^{-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

g

OC

.

(4 marks)

On its way to

C

P

passes through two small hoops

A

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

U

V

g

h

.

(3 marks)

It is given that, as

P

A

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

\alpha

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

.
(d) Determine an expression, in terms of

g

d

h

, for the speed with which

P

is projected from

O

●●●●●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}

t

seconds after it starts is given by

v = kt - 0.02t^2,

k

is a constant.

At the instant when

t = 15

the acceleration of the van is

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

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}

●●●●●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

t

v

^{-1}

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}

k

is a constant.
(a) Show that

k = 5

, explaining your method carefully.

(3 marks)
(b) Find the acceleration of

P

t = 2.5

(3 marks)

When

t = 0

P

passes through the fixed point

O

on the line.
(c) Find the distance of

P

O

t = 8

●●●●●Level 513 marksStart

Figure 2Diagram not accurately drawn.

Two small blocks,

A

B

, are at rest on a rough horizontal floor. Block

A

0.6

B

0.9

kg. The distance between

A

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}

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

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

A

B

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

A go-kart

P

starts from rest at

A

t = 0

and moves along the track towards

B

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

P

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}

t = 30

P

B

.

Given that the distance

AB

300

m,
(a) show that

V = 12

(3 marks)
(b) Find the acceleration of

P

t = 0

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}

t = 8

, a second go-kart

Q

starts from rest at

A

and moves along the track in the same direction as

P

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}

Q

catches up with

P

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

●●●●●Level 513 marksStart

Figure 1 (not accurately drawn)

A sledge

P

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

\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

\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

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

P

3m

Q

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

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

u

P

immediately after the collision.

(7 marks)

Given that

e = \frac12

(b) find, in terms of

m

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

Q

takes place at a distance

d

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

d

u

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

P

Q

●●●●●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

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

P

Q

3

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}

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

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

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

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

A

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

\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}

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}

\mathbf{j}

.

(2 marks)
(c) Find the velocity of

B

t = 0

, giving your answer in terms of

\mathbf{i}

\mathbf{j}

.

(2 marks)

At time

T_1

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

T_2 > 0

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

5T_2^2 + pT_2 + q = 0

p

q

are integers to be found.

(3 marks)

●●●●●Level 513 marksStart

Kinematics with calculus (a → v → s)

34 questions

LessonNot started

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

t \ge 1

, the position vector of a particle

P

\mathbf{r}

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

c

is a constant.

When

t = 3

, the bearing of

P

O

045^\circ

(a) Show that

c = 27

(3 marks)
(b) Find the speed of

P

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

●●●●●Level 511 marksStart

In this question the unit vectors

\mathbf{i}

\mathbf{j}

are in the directions east and north respectively.

A particle

P

1.2

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

\mathbf{F}

t

t \geq 0

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

P

, relative to a fixed origin

O

\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}

\mathbf{j}

, the acceleration of

P

t

.

(1 mark)
(c) Determine the magnitude of

\mathbf{F}

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

O

t = 6

●●●●●Level 513 marksStart

In this question the unit vectors

\mathbf{i}

\mathbf{j}

are in the directions east and north respectively.

A particle

P

2.5

kg moves on a smooth horizontal surface. At time

t

t \geq 0

, the only horizontal forces acting on

P

are a constant force

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

\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

t = 0

t = 5

●●●●●Level 513 marksStart

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

10

^{-2}

P

is thrown from a point

A

on a balcony with speed

15

^{-1}

at an angle of elevation

\theta

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

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

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

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

t

t \ge 0

v

^{-1}

Q

v = kt^2 + 1.2t

k

is a positive constant.
(d) Given that

P

Q

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}

t

seconds after it starts is given by

v = kt - 0.02t^2,

k

is a constant.

At the instant when

t = 15

the acceleration of the van is

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

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}

●●●●●Level 510 marksStart

P

moves in a straight line. The velocity

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

P

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,

k

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

P

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

k

t = 6

t = 10

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

(3 marks)
(b) Find the value of

t

0 \leqslant t \leqslant 10

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

t = 0

t = 10

.

Find the value of

k

●●●●●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

t

v

^{-1}

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}

k

is a constant.
(a) Show that

k = 5

, explaining your method carefully.

(3 marks)
(b) Find the acceleration of

P

t = 2.5

(3 marks)

When

t = 0

P

passes through the fixed point

O

on the line.
(c) Find the distance of

P

O

t = 8

●●●●●Level 513 marksStart

[In this question

\mathbf{i}

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

0.5

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

P

, so that at time

t

0 \le t \le 2

, the velocity of

P

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

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

t = 0

P

is at the fixed origin

O

.
(a) Show that the position vector of

P

O

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

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

P

O

●●●●●Level 514 marksStart

Motion in two dimensions with vectors

48 questions

LessonNot started

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

t \ge 1

, the position vector of a particle

P

\mathbf{r}

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

c

is a constant.

When

t = 3

, the bearing of

P

O

045^\circ

(a) Show that

c = 27

(3 marks)
(b) Find the speed of

P

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

●●●●●Level 511 marksStart

In this question the unit vectors

\mathbf{i}

\mathbf{j}

are in the directions east and north respectively.

A particle

P

1.2

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

\mathbf{F}

t

t \geq 0

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

P

, relative to a fixed origin

O

\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}

\mathbf{j}

, the acceleration of

P

t

.

(1 mark)
(c) Determine the magnitude of

\mathbf{F}

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

O

t = 6

●●●●●Level 513 marksStart

In this question the unit vectors

\mathbf{i}

\mathbf{j}

are in the directions east and north respectively.

A particle

P

2.5

kg moves on a smooth horizontal surface. At time

t

t \geq 0

, the only horizontal forces acting on

P

are a constant force

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

\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

t = 0

t = 5

●●●●●Level 513 marksStart

[In this question

\mathbf{i}

\mathbf{j}

are horizontal perpendicular unit vectors.]

A particle

P

0.5

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

P

, so that at time

t

0 \le t \le 2

, the velocity of

P

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

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

t = 0

P

is at the fixed origin

O

.
(a) Show that the position vector of

P

O

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

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

P

O

●●●●●Level 514 marksStart

[In this question

\mathbf{i}

\mathbf{j}

are perpendicular unit vectors, with

\mathbf{i}

\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

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

[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 $O$ .]

A patrol boat

A

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 $\tan\alpha = \dfrac{21}{20}$

The boats are modelled as particles.
(a) Find, in terms of

\mathbf{i}

\mathbf{j}

, the velocity of

B

\text{km h}^{-1}

(2 marks)

At 13:00

the position vector of $A$ is $(-4\mathbf{i} + 12\mathbf{j})$ km
the position vector of $B$ is $(5\mathbf{i} + 5\mathbf{j})$ km

t

hours after 13:00

the position vector of $A$ is $\mathbf{r}$ km, where $\mathbf{r} = (-4 + 36t)\mathbf{i} + 12\mathbf{j}$
the position vector of $B$ is $\mathbf{s}$ km

(b) Find an expression for

\mathbf{s}

t

\mathbf{i}

\mathbf{j}

.

(2 marks)
(c) Show that at time

t

hours after 13:00,

\overrightarrow{AB} = [(9 - 15t)\mathbf{i} + (20t - 7)\mathbf{j}]

km

(2 marks)
(d) Show that the boats will never collide.

(3 marks)
(e) Find the distance between the boats when

B

is north-east of

A

●●●●●Level 513 marksStart

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

Two drones,

A

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

A

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

\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}

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}

\mathbf{j}

.

(2 marks)
(c) Find the velocity of

B

t = 0

, giving your answer in terms of

\mathbf{i}

\mathbf{j}

.

(2 marks)

At time

T_1

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

T_2 > 0

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

5T_2^2 + pT_2 + q = 0

p

q

are integers to be found.

(3 marks)

●●●●●Level 513 marksStart

A particle starts at the point with position vector

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

m, with velocity

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

, and moves with constant acceleration

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

. Find
(a) its velocity after

2

s,
(b) its speed after

2

s,
(c) its position vector after

2

s, and (d) its distance from the origin

O

2

●●●●●Level 57 marksStart

A particle of mass

0.5

kg moves with velocity

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

^{-1}

t

seconds.

Find the magnitude of the resultant force on the particle when

t = 2

●●●●●Level 55 marksStart