A-level · Projectiles

Projectile motion formulaeExam-style questions

The standard results for a projectile launched at speed u and angle θ over level ground — time of flight 2u sinθ/g, greatest height u²sin²θ/(2g), range u²sin2θ/g — and using them in reverse to find u or θ.

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17 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
At time $t = 0$ , a particle leaves the point $O$ and then moves freely under gravity. Its initial velocity has magnitude $u$ m s $^{-1}$ and is directed upwards at an angle $\alpha$ to the horizontal. At time $t$ seconds the particle is at the point $(x, y)$ relative to $O$ , where $x$ metres is measured horizontally and $y$ metres is measured vertically upwards.
(a) Show that the equation of the path of the particle is

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

(6 marks)Figure 3 (not accurately drawn)

A ball is thrown from the point $A$ , which is $12.5$ m vertically above horizontal ground. The ball is thrown upwards at an angle of $\theta°$ to the horizontal, with speed $U$ m s $^{-1}$ , and lands on the ground at the point $B$ . The horizontal distance from $A$ to $B$ is $20$ m, as shown in Figure 3.

The ball is modelled as a particle moving freely under gravity. It is moving with speed $21$ m s $^{-1}$ at the instant it reaches $B$ .
(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 5 · 12 marksTry it
2
A projectile is launched at speed $u$ at angle $\alpha$ above the horizontal over level ground, reaching greatest height $H$ and range $R$ .

Show that $\dfrac{H}{R} = \dfrac{\tan\alpha}{4}$ . (4 marks)
Level 5 · 4 marksTry it
3
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 5 · 12 marksTry it