Overview:

1. Definitions and recap
2. Projecting a particle directly upwards
3. Projecting a particle at any angle
4. Projectile motion formulae

What is a projectile?

A projectile is an object that is projected or thrown. In this section, we consider the motion of a projectile in a two-dimensional vertical plane. In this section, we model the projectiles as particles. Whilst a particle can have mass, it has no volume and, crucial, no surface area. This means that we can ignore the effects of air resistance in our modelling. In this section, the projectiles have nothing to power them once projected—e.g. we might consider the trajectory or path of a ball, but we won’t be looking at a rocket whose engines are still generating thrust.

Constant acceleration formulae

You should be familiar with the constant acceleration or “suvat” formulae:

\(v=u+at\)

\(s=ut+\frac{1}{2}at^{2}\)

\(s=vt-\frac{1}{2}at^{2}\)

\(s=\frac{1}{2}\left(u+v\right)t\)

\(v^{2}=u^{2}+2as\)

Weight

Remember, the weight of an object is the gravitational force exerted on the object. On earth, an object of mass \(m\) kg experiences a force of \(mg \text{ N}\) directed towards the centre of the earth, where \(g \approx 9.8 \text{ ms}^{-2}\). The precise value of \(g\) varies depending on latitude and altitude. Note the units of \(g\): we have acceleration here. Any object that is under the influence of gravity alone (i.e. an object on which no other forces are being exerted) accelerates towards the centre of the earth at \(g \text{ ms}^{-2}\). The mass of the object is irrelevant. If you release a feather and a bowling ball from the same height above the ground, they should hit the ground at the same time. Typically this won’t happen, because these objects aren’t under the influence of gravity alone; the objects experience air resistance, which has a greater impact on the feather. In a vacuum, however, we see that the two objects do indeed fall at the same rate:

Splitting a velocity into perpendicular components

You should also be able to split a velocity into perpendicular components (most commonly into horizontal and vertical components), and also work backwards. In this applet, a velocity vector is shown in red. It can be split into horizontal and vertical components, shown as dashed vectors: