Resolve a launch speed U at angle \theta above horizontal.

ux=U\cos\theta and uy=U\sin\theta.

Vector form of position under constant acceleration?

\mathbf{r}=\mathbf{u}t+\frac12\mathbf{a}t^2 with \mathbf{a}=-g\mathbf{j}.

Condition for the highest point?

vy=0, so the velocity is horizontal.

Equation of the path of a projectile?

y=x\tan\theta-\frac{gx^2}{2U^2\cos^2\theta}, a parabola.

Stone thrown horizontally from height h: time to land?

h=\frac12gt^2, so t=\sqrt{\frac{2h}{g}}.

How does air resistance change real motion compared with the model?

Range and greatest height are reduced, and the path is no longer symmetric.

ProjectilesAQA A-Level Maths: Flashcards

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What are the modelling assumptions for a projectile?

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What are the modelling assumptions for a projectile?: Particle, no air resistance, constant acceleration $g$ downwards, motion in a vertical plane.
Resolve a launch speed $U$ at angle $\theta$ above horizontal.: $u_x=U\cos\theta$ and $u_y=U\sin\theta$ .
What is the horizontal acceleration of a projectile?: Zero, so $v_x$ is constant.
What is the vertical acceleration of a projectile (up positive)?: $-g=-9.8\ \text{m s}^{-2}$ .
Vector form of position under constant acceleration?: $\mathbf{r}=\mathbf{u}t+\frac12\mathbf{a}t^2$ with $\mathbf{a}=-g\mathbf{j}$ .
Condition for the highest point?: $v_y=0$ , so the velocity is horizontal.
Greatest height formula (launch from ground)?: $H=\frac{U^2\sin^2\theta}{2g}$ .
Time of flight formula (same level launch and landing)?: $T=\frac{2U\sin\theta}{g}$ .
Range formula (same level)?: $R=\frac{U^2\sin2\theta}{g}$ .
Speed at a point where components are $v_x$ and $v_y$ ?: $\sqrt{v_x^2+v_y^2}$ .
Equation of the path of a projectile?: $y=x\tan\theta-\frac{gx^2}{2U^2\cos^2\theta}$ , a parabola.
Stone thrown horizontally from height $h$ : time to land?: $h=\frac12gt^2$ , so $t=\sqrt{\frac{2h}{g}}$ .
How does air resistance change real motion compared with the model?: Range and greatest height are reduced, and the path is no longer symmetric.

Exam questions on Projectiles

A ball is projected from a point $O$ on level ground with speed $20\ \text{m s}^{-1}$ at $30^\circ$ above the horizontal. Model the ball as a particle moving freely under gravity, with $g=9.8\ \text{m s}^{-2}$ .
Calculate the horizontal distance the ball travels before landing.2 marks
A stone is thrown horizontally at $15\ \text{m s}^{-1}$ from a window $19.6$ m above level ground. Model the stone as a particle moving freely under gravity, with $g=9.8\ \text{m s}^{-2}$ .
Find the speed of the stone as it hits the ground.2 marks
A particle is projected from a point $O$ on horizontal ground with initial velocity $(6\mathbf{i}+14.7\mathbf{j})\ \text{m s}^{-1}$ , where $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertically upward unit vectors. The particle moves freely under gravity, with $g=9.8\ \text{m s}^{-2}$ .
Find the velocity of the particle after $2$ s, and state whether it is moving upwards or downwards at that time.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).