Projectile Motion
Any object which is allowed to experience the effect of gravity while it is in flight is categorised
as a projectile. Projectiles are acted on by the force of gravity and hence their flight represents
the effect of this force.
The object is projected at an angle, θ, above the horizontal with an initial velocity, u. As soon as
it is released, the gravitational force begins to affect it. This force reduces the vertical component
(u sinθ) of the velocity until it has a magnitude of zero at the apex (maximum height) of the
flight. At this point the object’s velocity still has a horizontal component. After the object passes
its maximum height, it begins to descend as the gravitational force continues to act. The vertical
component is now acting downwards. The horizontal component remains constant for the
entire journey.
The equations of motion must be modified to compensate for the gravitational force.
Displacement
Horizontally
There are no resistive forces in the horizontal direction. Hence the velocity in this direction is
constant. Therefore the displacement in the x-direction is given as:
Vertically
Gravity acts in the vertical direction and the displacement in this plane is given by:
Velocity at any time
Recall:
Horizontally
The horizontal component of velocity remains constant.
Vertically
Time taken to reach the apex (maximum height)
At the apex, vy is zero. Therefore
Time taken to land
This is the time taken for the projectile to travel its entire path. At the end of the flight the
displacement in the ‘y’ direction is zero. (sy =0). Hence:
This reduces to:
This is twice as much time as it takes to reach the apex of flight.
Range
The range is the maximum distance travelled in the horizontal direction. Recall:
Additionally, the total time of flight is given as:
Substituting this equation into the equation for displacement in the horizontal direction provides
an equation for determining the range of a projectile.
Now
(this is known as a double angle formula)
Therefore:
Maximum Height
At the apex
. The initial vertical velocity is given by
Hence the equation for the maximum height is given by:
, and