4.1.12

# Projectile Motion

Test yourself

## Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. We assume air resistance is negligible.

### Horizontal and vertical

• The most important thing to remember about projectile motion is that the horizontal and vertical components of the problem are independent.
• This means we can solve two sets of equations, one for each direction.

### Components of displacement

• This diagram shows the displacement, s, of a football at a point along its path (trajectory).
• The displacement vector has components sx (along the horizontal axis) and sy (along the vertical axis).
• Its magnitude is s, and it makes an angle θ with the horizontal.
• The vertical component is sy = s sinθ.
• The horizontal component is sx = s cosθ.

### Components of acceleration

• Projectiles are only acted upon by one force: gravity.
• Therefore, the vertical component of acceleration, ay, is equal to the acceleration due to gravity.
• ay = g = −9.81m/s².
• Remember the minus sign!
• There is no force acting in the horizontal direction, so the horizontal component of acceleration, ax, is zero.
• ax = 0.

### Components of velocity

• Velocity can also be separated into components.
• For a projectile travelling at an angle θ at a velocity v:
• vy = v sinθ
• vx = v cosθ

## Projectile Motion - Worked Example

The following steps can be used to solve problems with projectile motion.

### Step 1 - resolve components

• Find the horizontal information:
• sx, ux, vx and ax (this will be zero).
• Find the vertical information:
• sy, uy, vy and ay (this will be g = -9.81m/s²).
• Time, t, is the same in each component.

### Step 2 - choose equations for each part

• Choose which of the 'suvat' equations you need to use for each component of motion.
• Remember the equation you need for the vertical part may be different to the one you need for the horizontal.

### Step 3 - solve each component of motion

• Use the 'suvat' equation you've identified to solve each component of motion.
• Remember to solve them separately.

### Step 4 - recombine the variables

• If you need to, you can find the total velocity vector, displacement vector and angle by recombining the components of the vector.
• $s = \sqrt{s_x^2 + s_y^2}$
• $v = \sqrt{v_x^2 + v_y^2}$
• $\theta = \tan^{-1}(\frac{v_x}{v_y}) = \tan^{-1}(\frac{s_x}{s_y})$