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DifficultyAll Level 2 Level 3 Level 4 Level 5

Projection at an angle

44 questions

LessonNot started

O

on horizontal ground and first hits the ground again at the point

A

, where

OA = 30

m.

In an initial model

the ball is modelled as a particle moving freely under gravity
the ball is projected from $O$ with speed $U\ \text{m s}^{-1}$ at an angle $\alpha$ to the horizontal, where $0 < \alpha < 90^\circ$

Using the model,
(a) show that

U^2\sin\alpha\cos\alpha = 147

(6 marks)

Using the model, the greatest height of the ball above the ground is

10

m.
(b) Show that

U^2 = 306.25

(4 marks)

In a refinement to the model, the effect of air resistance is included. The motion of the ball, from

O

A

, is now modelled as that of a particle projected at the same angle

\alpha

whose initial speed is

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

. This refined model is used to calculate a value for

V

.
(c) State which is greater,

U

V

, giving a reason for your answer.

(1 mark)
(d) State one further refinement to the model that would make the model more realistic.

(1 mark)

●●●●●Level 512 marksStart

A small ball is projected with speed

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

from a point

O

on horizontal ground in a park.

After moving for

T

seconds, the ball passes through the point

A

, which is the top of a tree.

The point

A

8

m horizontally and

8

m vertically from the point

O

.

The motion of the ball from

O

A

is modelled as that of a particle moving freely under gravity.

Given that the ball is projected at an angle

\alpha

to the horizontal, use the model to
(a) show that

T = \dfrac{4}{7\cos\alpha}

(2 marks)
(b) show that

\tan^2\alpha - 5\tan\alpha + 6 = 0

(5 marks)
(c) find the greatest possible height of the ball above the ground.

(3 marks)

The model does not include air resistance.
(d) State one other limitation of the model.

(1 mark)

●●●●●Level 511 marksStart

A small pebble is lying on horizontal ground. The pebble is struck by the blade of a lawnmower and is projected from the point

O

where it was lying.

In an initial model

the pebble is modelled as a particle $P$ moving freely under gravity
the pebble is projected from $O$ with speed $21\ \text{m s}^{-1}$ at an angle $\alpha$ to the horizontal, where $\tan\alpha = \frac{1}{3}$
the pebble hits the ground at the point $A$

For the motion of

P

from

O

A

at time $t$ seconds, the horizontal distance of $P$ from $O$ is $x$ metres
at time $t$ seconds, the vertical distance of $P$ above the ground is $y$ metres

(a) Using the model, show that

y = \frac{x}{3} - \frac{x^2}{81}

(6 marks)
(b) Use the answer to part (a), or otherwise, to find the length

OA

.

(2 marks)

Using the model, the greatest height of the pebble above the ground is found to be

H

metres.
(c) Use the answer to part (a), or otherwise, to find the value of

H

.

(2 marks)

The model is refined to include air resistance. Using this refined model, the greatest height of the pebble above the ground is found to be

K

metres.
(d) State which is greater,

H

K

, justifying your answer.

(1 mark)
(e) State one limitation of this refined model.

(1 mark)

●●●●●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

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 particle

P

is projected with speed

u

m s

^{-1}

from a fixed point

O

, at an angle

\alpha

above the horizontal, and moves freely under gravity. When

P

has moved a horizontal distance of

x

metres from

O

, it is at a height of

y

metres above

O

.
(a) Show that

y = x\tan\alpha - \frac{gx^2}{2u^2}\left(1 + \tan^2\alpha\right)

(6 marks)Figure 3 (not accurately drawn)

A stone is thrown from a point

A

with speed

U

m s

^{-1}

\theta°

above the horizontal. The point

B

is on horizontal ground, vertically below

A

, with

AB = 15

m. The stone lands on the ground at the point

C

, where

BC = 21

m, as shown in Figure 3.

The stone is modelled as a particle moving freely under gravity. Just before it lands at

C

, the speed of the stone is

23

m s

^{-1}

.
(b) Use the principle of conservation of mechanical energy to find the value of

U

.

(3 marks)
(c) Find the value of

\theta

.

(3 marks)

●●●●●Level 512 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 ball is projected at

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

25^\circ

above the horizontal over level ground.
(a) Find the time of flight.
(b) Find the horizontal range. Give answers to 3 s.f. (

g = 9.8

●●●●●Level 54 marksStart

A stone is projected at

25

m s

^{-1}

40^{\circ}

above the horizontal.

Find the speed and direction of motion of the stone

2

seconds after projection. (6 marks)

●●●●●Level 56 marksStart

A ball is kicked from level ground at

22

m s

^{-1}

35^{\circ}

above the horizontal, directly towards a wall

15

m away. The wall is

3

m tall.

Determine whether the ball clears the wall, and by what vertical margin. (5 marks)

●●●●●Level 55 marksStart

A particle is projected from a point on horizontal ground with speed

u

at an angle

\theta

above the horizontal, and moves freely under gravity until it returns to the ground.
(a) Show that the horizontal range is

R = \dfrac{u^2\sin 2\theta}{g}

. (5 marks)
(b) Given that

u = 28

m s⁻¹ and

R = 60

m, find the two possible values of

\theta

, to 1 decimal place. Take

g = 9.8

m s⁻². (4 marks)
(c) State which of your two angles gives the longer time of flight, justifying your answer. (2 marks)
(d) State one assumption of the model. (1 mark)

●●●●●Level 512 marksStart

A ball is kicked from a point

O

on horizontal ground with speed

14

m s⁻¹ at an angle

\theta

above the horizontal, where

0 < \theta < 90^\circ

. The ball is modelled as a particle moving freely under gravity. Take

g = 9.8

m s⁻².

After

t

seconds the horizontal and vertical displacements of the ball from

O

are

x

metres and

y

metres respectively.
(a) Show that

y = x\tan\theta - \dfrac{x^2}{40}\left(1 + \tan^2\theta\right)

. (3 marks)

The ball passes through the point

P

, which is

10

m horizontally from

O

and

5

m above the ground.
(b) Show that

\tan^2\theta - 4\tan\theta + 3 = 0

, and hence find the two possible values of

\theta

. (4 marks)
(c) Show that, for either value of

\theta

, the speed of the ball as it passes through

P

7\sqrt{2}

m s⁻¹. (3 marks)
(d) For the smaller value of

\theta

, determine whether the ball is rising, falling or at its highest point as it passes through

P

. Fully justify your answer. (2 marks)

●●●●●Level 512 marksStart

A stone is thrown from a point

O

at the top of a vertical cliff,

25

m above horizontal ground. It is thrown with speed

15

m s⁻¹ at an angle of

30^\circ

above the horizontal.

The stone is modelled as a particle moving freely under gravity. Take

g = 9.8

m s⁻².
(a) Find the time taken for the stone to reach the ground, giving your answer to

3

significant figures. (5 marks)
(b) Find the horizontal distance from the foot of the cliff at which the stone lands, giving your answer to

3

significant figures. (2 marks)
(c) Find the speed of the stone as it hits the ground, giving your answer to

3

significant figures. (3 marks)
(d) Find the greatest height above the ground reached by the stone, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 marksStart

A ball is kicked from a point

O

on horizontal ground with speed

20

m s⁻¹ at an angle

\alpha

above the horizontal, where

\tan\alpha = \dfrac34

.

The ball is modelled as a particle moving freely under gravity. Take

g = 9.8

m s⁻².
(a) Find the time of flight of the ball, giving your answer to

3

significant figures. (3 marks)
(b) Find the horizontal distance from

O

at which the ball lands, giving your answer to

3

significant figures. (2 marks)
(c) A wall of height

5

m stands on the ground at a horizontal distance of

30

m from

O

, in the plane of the ball's motion.

Determine whether the ball passes over the wall. (5 marks)
(d) State one assumption made in the model, and explain how your answer to part (c) might change if this assumption were not made. (3 marks)

●●●●●Level 513 marksStart

A ball is struck from a point

O

on horizontal ground with speed

21

m s⁻¹ at an angle

\theta

above the horizontal. The ball is modelled as a particle moving freely under gravity. Taking

O

as the origin,

x

metres and

y

metres are the horizontal and vertical displacements of the ball from

O

at time

t

seconds. In this question use

g = 9.8

m s⁻².
(a) Show that the equation of the trajectory of the ball is

y = x\tan\theta - \dfrac{x^2}{90}\left(1 + \tan^2\theta\right)

(4 marks)

The ball passes through the point

Q

, which is

30

m horizontally from

O

and

2.5

m above the ground.
(b) Find the two possible values of

\tan\theta

. (4 marks)
(c) Find the speed of the ball as it passes through

Q

, giving your answer in the form

k\sqrt{2}

m s⁻¹, and explain why this speed is the same for both possible values of

\tan\theta

. (5 marks)

●●●●●Level 513 marksStart

Projectile motion formulae

41 questions

LessonNot started

A football is kicked from a point

O

on horizontal ground and first hits the ground again at the point

A

, where

OA = 30

m.

In an initial model

the ball is modelled as a particle moving freely under gravity
the ball is projected from $O$ with speed $U\ \text{m s}^{-1}$ at an angle $\alpha$ to the horizontal, where $0 < \alpha < 90^\circ$

Using the model,
(a) show that

U^2\sin\alpha\cos\alpha = 147

(6 marks)

Using the model, the greatest height of the ball above the ground is

10

m.
(b) Show that

U^2 = 306.25

(4 marks)

In a refinement to the model, the effect of air resistance is included. The motion of the ball, from

O

A

, is now modelled as that of a particle projected at the same angle

\alpha

whose initial speed is

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

. This refined model is used to calculate a value for

V

.
(c) State which is greater,

U

V

, giving a reason for your answer.

(1 mark)
(d) State one further refinement to the model that would make the model more realistic.

(1 mark)

●●●●●Level 512 marksStart

A small pebble is lying on horizontal ground. The pebble is struck by the blade of a lawnmower and is projected from the point

O

where it was lying.

In an initial model

the pebble is modelled as a particle $P$ moving freely under gravity
the pebble is projected from $O$ with speed $21\ \text{m s}^{-1}$ at an angle $\alpha$ to the horizontal, where $\tan\alpha = \frac{1}{3}$
the pebble hits the ground at the point $A$

For the motion of

P

from

O

A

at time $t$ seconds, the horizontal distance of $P$ from $O$ is $x$ metres
at time $t$ seconds, the vertical distance of $P$ above the ground is $y$ metres