Overview of Projectile Motion (HSC SSCE Physics): Revision Notes

Overview of Projectile Motion

What is a projectile?

A projectile is any object that has been launched, thrown, or fired into the air. Once in motion, a projectile doesn't power itself - unlike a bird or aeroplane that uses energy to maintain flight. Common examples include:

• A kicked football
• A bullet fired from a gun
• A thrown basketball
• Water droplets shot by an archer fish

After launch, a projectile's motion depends only on two factors: its initial velocity (the speed and direction at launch) and the gravitational force pulling it downward.

Understanding trajectory

The trajectory is the path that a projectile follows through the air. Two key elements determine this path:

1. The initial velocity - how fast and in what direction the projectile starts moving
2. The forces acting on it - in our simplified model, only gravity

Simplifying assumptions

To make calculations manageable, we assume that air resistance has no significant effect. This means gravity is the only force we need to consider after launch.

The gravitational force near Earth's surface is constant and always points vertically downward, with magnitude:

$F = mg$

where $m$ is the projectile's mass and $g$ is the acceleration due to gravity.

Acceleration of a projectile

Using Newton's second law, we can find the acceleration of any projectile:

$a = \frac{F}{m} = \frac{mg}{m} = g$

This shows that every projectile accelerates downward at the same rate, regardless of its mass. We use the value:

$g = -9.8 \, \text{m} \cdot \text{s}^{-2}$

The negative sign indicates downward direction.

Breaking acceleration into components

Since acceleration is a vector, we can separate it into perpendicular components:

Breaking down initial velocity

When a projectile launches at an angle, we need to split its initial velocity into horizontal and vertical parts. The angle above the horizontal is called the launch angle, represented by $\theta$.

Using trigonometry, the initial velocity components are:

• Horizontal component: $u_x = u \cos\theta$
• Vertical component: $u_y = u \sin\theta$

where $u$ is the magnitude (size) of the initial velocity.

Velocity during flight

The general equation for velocity under constant acceleration is:

$\vec{v} = \vec{u} + \vec{a}t$

We can separate this into horizontal and vertical components.

Horizontal velocity

For the horizontal direction:

$v_x = u_x + a_x t$

Since $a_x = 0$ (no horizontal acceleration):

$v_x = u_x$

Vertical velocity

For the vertical direction:

$v_y = u_y + a_y t = u_y + gt$

Position of a projectile

The position equation for constant acceleration is:

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

Position has both horizontal ($x$) and vertical ($y$) components.

Horizontal position

$x = x_0 + u_x t + \frac{1}{2}a_x t^2$

Since $a_x = 0$:

$x = x_0 + u_x t$

If we set the launch point as the origin ($x_0 = 0$):

$x = u_x t$

Vertical position

$y = y_0 + u_y t + \frac{1}{2}a_y t^2$

Substituting $a_y = g$:

$y = y_0 + u_y t + \frac{1}{2}gt^2$

If the launch point is the origin ($y_0 = 0$):

$y = u_y t + \frac{1}{2}gt^2$

Understanding maximum height

When the vertical velocity reaches zero, the projectile stops rising and begins to fall. This point represents the maximum height of the trajectory.

Notice that at maximum height:

• The vertical velocity is zero
• The horizontal velocity remains unchanged
• The projectile continues moving horizontally

The trajectory is symmetric - the path after reaching maximum height mirrors the path before it.

Summary of projectile motion equations

The table below summarises all the key equations for analysing projectile motion:

Component	Horizontal	Vertical
Position	$x = u_x t = (u \cos\theta)t$	$y = y_0 + u_y t + \frac{1}{2}gt^2 = y_0 + (u \sin\theta)t + \frac{1}{2}gt^2$
Velocity	$v_x = u_x = u \cos\theta$	$v_y = u_y + gt = u \sin\theta + gt$
Acceleration	$a_x = 0$	$a_y = g = -9.8 \, \text{m} \cdot \text{s}^{-2}$