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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 by a catapult.

In an initial model

the stone is modelled as a particle moving freely under gravity
the stone is projected from $O$ with speed $24.5\ \text{m s}^{-1}$ at an angle $\alpha$ to the horizontal, where $\tan\alpha = \frac{1}{2}$
the stone hits the ground at the point $A$

For the motion of the stone from

O

A

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

Using the model,
(a) show that

y = \frac{x}{2} - \frac{x^2}{98}

(5 marks)
(b) Use the answer to part (a), or otherwise, to find the distance

OA

.

(2 marks)

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

(1 mark)

●●●●●Level 48 marksStart

At a science fair, a small ball is projected from a point

O

on horizontal ground by a compressed-air launcher.

The ball is projected with speed

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

at an angle

\alpha

to the horizontal, where

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

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

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

7.2

m.
(a) Show that

U^2 = 392

(3 marks)
(b) Find the time taken for the ball to return to the ground.

(2 marks)
(c) Find the horizontal distance travelled by the ball in this time.

(2 marks)
(d) State one limitation of the model that could affect the answer to part (c).

(1 mark)

●●●●●Level 48 marksStart

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

10

m s

^{-2}

.

A stone is thrown from a point

A

at the top of a harbour wall with a speed of

18

m s

^{-1}

at an angle of

35°

above the horizontal. While it is in the air, the stone is modelled as a particle moving freely under gravity. The stone first lands at a point

B

on the beach, which is

6.5

m below the level of

A

(see diagram).(a) Determine the time taken for the stone to travel from

A

B

.

(3 marks)
(b) Determine the horizontal distance of

B

from

A

.

(2 marks)
(c) Determine the direction of motion of the stone

2

seconds after it is thrown.

(4 marks)

In fact, the horizontal distance from

A

B

is found to be less than the answer to part (b).
(d) State one factor that could account for this difference.

(1 mark)

●●●●●Level 410 marksStart

A small ball

K

is kicked from a point

O

on horizontal ground. Axes

Ox

, horizontal along the ground, and

Oy

, vertically upwards, are fixed in the vertical plane of the motion; distances are in metres.

K

is kicked with a velocity whose components along

Ox

and

Oy

are

U

m s

^{-1}

and

V

m s

^{-1}

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

K

has equation

2U^2y = 2UVx - gx^2

.

(3 marks)

K

just passes over the top of a vertical flagpole of height

a

m, which stands at a horizontal distance

2a

m from

O

K

first strikes the ground at a horizontal distance

a

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

K

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

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

K

20

m s

^{-1}

, determine the value of

a

.

(2 marks)

●●●●●Level 410 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

[In this question

\mathbf{i}

and

\mathbf{j}

are unit vectors, with

\mathbf{i}

horizontal and

\mathbf{j}

vertical.]Figure 2 (not accurately drawn)

A point

A

lies on horizontal ground. At time

t = 0

, a particle

P

is projected from

A

with velocity

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

m s

^{-1}

. The particle moves freely under gravity and first hits the ground at the point

B

, as shown in Figure 2.

Particle

P

reaches its greatest height above the ground at time

T_1

seconds.
(a) Find the value of

T_1

(2 marks)

At the same instant that

P

is projected, a second particle

Q

is projected from

A

with velocity

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

m s

^{-1}

. Particle

Q

also moves freely under gravity.
(b) Find, to 2 significant figures, the vertical distance between

P

and

Q

at time

T_1

seconds.

(3 marks)

When

P

reaches

B

Q

is still in the air and is moving in a direction

\alpha°

below the horizontal.
(c) Find the value of

\alpha

.

(4 marks)

At time

T_2

seconds,

Q

is moving perpendicular to the direction in which it was projected.
(d) Find the value of

T_2

(3 marks)

●●●●●Level 412 marksStart

A stone is projected at

25

m s

^{-1}

40^{\circ}

above the horizontal from level ground.

Find the height of the stone when its horizontal displacement is

20

m. (5 marks)

●●●●●Level 45 marksStart

A ball is projected from a point

O

at the top of a vertical cliff,

20

m above the sea. The ball is projected with speed

28

m s⁻¹ at an angle

\alpha

above the horizontal, where

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

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

g = 9.8

m s⁻².(a) Find the time taken for the ball to hit the sea, explaining why one root of your equation is rejected. (5 marks)
(b) Find the horizontal distance from the foot of the cliff to the point where the ball lands. (2 marks)
(c) Find the speed of the ball as it hits the sea, giving your answer in the form

k\sqrt{6}

m s⁻¹. (3 marks)
(d) Suggest two refinements to the model that would make it more realistic. (2 marks)

●●●●●Level 412 marksStart

A football is kicked from a point

O

on level ground with speed

20

m s⁻¹ at an angle

\alpha

above the horizontal, where

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

. A vertical fence of height

2.5

m stands

24

m from

O

, perpendicular to the plane of the ball's motion. Air resistance is neglected. Take

g = 9.8

m s⁻².
(a) Show that the ball reaches the plane of the fence

1.5

seconds after being kicked. (2 marks)
(b) Determine whether the ball passes over the fence and, if so, the height by which it clears the fence, giving your answer to 3 significant figures. (3 marks)
(c) Find the speed of the ball as it crosses the plane of the fence. (3 marks)
(d) State two assumptions, other than neglecting air resistance, made in modelling the motion of the ball. (2 marks)

●●●●●Level 410 marksStart

A ball is projected from a point

O

on horizontal ground with speed

14

m s⁻¹ at an angle

\theta

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

g = 9.8

m s⁻².

At time

t

seconds after projection the ball is

x

m horizontally from

O

and

y

m above the ground.
(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

Q

, which is

8

m horizontally from

O

and

2

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

\tan\theta

. (3 marks)
(c) For the smaller of the two possible angles of projection, determine the direction in which the ball is moving as it passes through

Q

. (2 marks)

●●●●●Level 48 marksStart

In this question use

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

A football is kicked from a point on horizontal ground with an initial velocity of

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

at an angle

\theta

above the horizontal.

The football reaches a maximum vertical height of

h

metres above the ground.
(a) Show that

h = 10\sin^2\theta

(3 marks)
(b) Hence, given that

0^\circ \le \theta \le 50^\circ

, find the maximum value of

h

.

(2 marks)
(c) Ffion claims that the mass of the football changes the maximum vertical height.

State whether Ffion is correct, giving a reason for your answer.

(1 mark)

●●●●●Level 46 marksStart

In this question use

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

A golf ball is struck from a point on horizontal ground with an initial velocity of

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

at an angle

\theta

above the horizontal.

The golf ball reaches a maximum vertical height of

h

metres above the ground.
(a) Show that

h = 22.5\sin^2\theta

(3 marks)
(b) Hence, given that

0^\circ \le \theta \le 35^\circ

, find the maximum value of

h

.

(2 marks)
(c) Kofi claims that the air resistance acting on the golf ball changes the maximum vertical height.

State whether Kofi is correct, giving a reason for your answer.

(1 mark)

●●●●●Level 46 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

Projectile motion formulae

41 questions

LessonNot started

A small stone is projected from a point

O

on horizontal ground by a catapult.

In an initial model

the stone is modelled as a particle moving freely under gravity
the stone is projected from $O$ with speed $24.5\ \text{m s}^{-1}$ at an angle $\alpha$ to the horizontal, where $\tan\alpha = \frac{1}{2}$
the stone hits the ground at the point $A$

For the motion of the stone from

O

A

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