A projectile is a body that falls freely under gravity ie the only force acting on it is gravity. In fact this is never strictly true, since there is always air resistance for example, but these other forces are small compared to gravity in most circumstances so we ignore them.
We can resolve the initial velocity of the ball vertically and horizontally as shown:
The horizontal velocity is constant since there are no horizontal forces acting – gravity acts vertically. Horizontally therefore
Vertically however, the particle is accelerating downwards. We usually take this acceleration as negative and equal to Vertically, we can apply all the normal suvat equations:
becomeswhich we have already.
We can also derive an equation between and by rearranging (1) to give and substituting this into (3):
We can find the maximum height usingwithobtaining
We can find the range by substitutingin (5) to getSincethe maximum range is given bywhen