A-level · Projectiles

Projection at an angleExam-style questions

Resolving the launch velocity into u cosθ (horizontal, constant) and u sinθ (vertical, under gravity), then using suvat in each direction to find the time of flight, greatest height and range.

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40 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
[In this question $\mathbf{i}$ and $\mathbf{j}$ are unit vectors, with $\mathbf{i}$ horizontal and $\mathbf{j}$ vertical.]Figure 1 (not accurately drawn)

A small stone is thrown from a point $A$ on horizontal ground. The stone leaves $A$ with velocity $(6\mathbf{i} + 13\mathbf{j})$ m s $^{-1}$ and lands at the point $B$ , also on the ground, as shown in Figure 1.

The stone is modelled as a particle moving freely under gravity.
(a) Find the distance $AB$ .

(4 marks)

While the stone is in the air, there is a period of $T$ seconds during which its speed is at most $10$ m s $^{-1}$ .
(b) Find the value of $T$ .

(3 marks)

At a certain instant, the velocity of the stone is perpendicular to its velocity as it left $A$ . At this instant the stone is $h$ metres above the ground.
(c) Find the value of $h$ .

(4 marks)
Level 4 · 11 marksTry it
2
[In this question $\mathbf{i}$ and $\mathbf{j}$ are unit vectors, with $\mathbf{i}$ horizontal and $\mathbf{j}$ vertical.]Figure 2 (not accurately drawn)

Two particles, $P$ and $Q$ , are projected at the same instant from the point $O$ on horizontal ground. The initial velocity of $P$ is $(7\mathbf{i} + 5\mathbf{j})$ m s $^{-1}$ and the initial velocity of $Q$ is $(3\mathbf{i} + 8\mathbf{j})$ m s $^{-1}$ . Each particle moves freely under gravity.

Figure 2 shows the path of $P$ , which lands on the ground at the point $A$ .

Particle $P$ reaches its greatest height $T_1$ seconds after projection.
(a) Find the value of $T_1$

(2 marks)
(b) Find the height of $Q$ above $P$ at this instant, giving your answer to 2 significant figures.

(3 marks)

When $P$ lands at $A$ , $Q$ is still in the air and is descending at $\alpha°$ to the horizontal.
(c) Find the value of $\alpha$ .

(4 marks)

$T_2$ seconds after projection, $Q$ is moving at right angles to its direction of projection.
(d) Find the value of $T_2$

(3 marks)
Level 4 · 12 marksTry it
3
A stone is projected at $25$ m s $^{-1}$ at $40^{\circ}$ above the horizontal from level ground.

Find the height of the stone when its horizontal displacement is $20$ m. (5 marks)
Level 4 · 5 marksTry it
4
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 4 · 12 marksTry it
5
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 4 · 10 marksTry it

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