Think about horizontal and vertical motion separately

• A projectile is any object with an initial velocity that then moves freely under gravity
• The horizontal and vertical components of a projectile's motion are independent
• The horizontal velocity remains constant while the vertical velocity changes due to gravity
• This causes the curved path typical of projectile motion
• The vertical component of velocity is 0 m/s at the highest point

Worked example - Calculating the range of a projectile

Sharon fires a scale model horizontally from 15 m high with a speed of 100 m/s. Determine the time of flight and horizontal distance traveled, assuming no air resistance.

Step 1: Formula

$\text{s = u t + }\frac{1}{2}\text{a t}^2$

Step 2: Rearrangement

$\text{t = }\sqrt\frac{\text{2 s}}{\text{a}}$

Step 3: Substitution and correct evaluation

s = 15 m

u = 0 m s$^{-1}$

v =

a = 9.81 m s$^{-2}$

t = ?

$\text{t = }\sqrt\frac{2\times15}{9.81}$

t = 1.7487 s

Step 4: Calculate range

range = horizontal velocity x flight time

range = 100 x 1.7487 = 174.9 m

Motion at an angle requires vector resolving

• Forces and velocities can act in any direction
• Motion must be resolved into horizontal and vertical vectors using trigonometry
• The initial velocity determines the time of flight and maximum height

Consider an object moving with velocity R at an angle of $\theta$ to the horizontal.

The horizontal and vertical components can be calculated:

• $\text{horizontal component = R} \cos(\theta)$
• $\text{vertical component = R} \sin(\theta)$

Worked example - Javelin throw

A javelin is thrown at a speed of 21 m/s at an angle of 45° to the horizontal. Determine the range of the javelin.

Step 1: Calculate vertical and horizontal components

$u_x = \cos(45°) \times 21 = 14.849 \text{ m s}^{-1}$

$u_y = \sin(45°) \times 21 = 14.849\text{ m s}^{-1}$

Step 2: Calculate time to reach highest point

s = 0 m

u_y = 14.849 m s$^{-1}$

v_y = 0 m s$^{-1}$

a = - 9.81 m s$^{-2}$

t = ?

$t = \frac{\text{v-u}}{\text{a}}$

$t = \frac{0 - 14.85}{-9.81} = 1.5136 \text{ s}$

Step 3: Calculate total flight time

total flight time = 2 x 1.5136 = 3.027 s

Step 4: Calculate range of javelin

range = horizontal velocity x flight time

range = 14.849 x 3.027 = 44.95 m