Projectile motion

Specifying velocity by speed and angle or by components, deriving the equations of motion of a projectile from the acceleration vector, time of flight, greatest height, range, and the velocity at any time.

Setting up

A projectile launched with speed $V$ at angle $\theta$ above the horizontal has initial velocity $\begin{pmatrix} V\cos\theta \\ V\sin\theta \end{pmatrix}$. Ignoring air resistance, the only acceleration is gravity: $\underset{\sim}{a} = \begin{pmatrix} 0 \\ -g \end{pmatrix}$.

Equations of motion (launched from the origin)

$$\underset{\sim}{v} = \begin{pmatrix} V\cos\theta \\ V\sin\theta - gt \end{pmatrix} \qquad \underset{\sim}{r} = \begin{pmatrix} Vt\cos\theta \\ Vt\sin\theta - \tfrac12 gt^2 \end{pmatrix}$$

These come from integrating $\underset{\sim}{a}$ twice, as in the last section. In exams you are expected to derive them, not just quote them. The horizontal velocity never changes.

Key quantities

Example 1●●

A ball is kicked at $20$ m/s at $30^\circ$ to the horizontal. Take $g = 10$ m/s$^2$. Find the time of flight, the greatest height and the range.

$\underset{\sim}{r} = \begin{pmatrix} 10\sqrt3\,t \\ 10t - 5t^2 \end{pmatrix}$.

It lands when $10t - 5t^2 = 0$, so the time of flight is $2$ s.

The vertical velocity $10 - 10t$ is zero at $t = 1$, so the greatest height is $10 - 5 = 5$ m.

The range is $10\sqrt3 \times 2 = 20\sqrt3 \approx 34.64$ m.

Example 2: velocity at a given time●●●

For the same ball, find its speed and direction at $t = 1.5$ s.

$\underset{\sim}{v} = \begin{pmatrix} 10\sqrt3 \\ 10 - 15 \end{pmatrix} = \begin{pmatrix} 10\sqrt3 \\ -5 \end{pmatrix}$.

Speed $= \sqrt{300 + 25} = 5\sqrt{13} \approx 18.03$ m/s.

$\tan\alpha = \dfrac{5}{10\sqrt3}$, so it is moving at about $16.1^\circ$ below the horizontal.

Flashcards

Initial velocity at speed $V$, angle $ heta$

$\begin{pmatrix} V\cos\theta \\ V\sin\theta \end{pmatrix}$

Greatest height

When the vertical velocity is $0$.

Time of flight

When $y = 0$ again (for level ground).

Horizontal velocity of a projectile

Constant: $V\cos\theta$.

Practice

A ball is kicked at $20$ m/s at $30^\circ$. Take $g = 10$. Find the time of flight in seconds.
Find the greatest height of that ball, in metres.
Find the range of that ball, in metres, correct to two decimal places.
Find the speed of that ball at $t = 1.5$ s, correct to two decimal places.
Find the horizontal component of that ball's velocity. Give an exact answer or a decimal.
A stone is thrown at $10$ m/s horizontally from the top of a cliff $45$ m high. Take $g = 10$. How far from the base of the cliff does it land, in metres?
For a projectile with no air resistance, the horizontal velocity:
At the greatest height, the velocity is:
The acceleration of a projectile is: