## 3. What goes up must come down - the flight of projectiles

When a projectile is fired vertically, it has an initial upwards velocity as it leaves the muzzle of the gun. The downwards attraction of gravity slows the projectile, until it reaches a height when the upwards velocity is zero. The projectile then falls back to earth, accelerating under gravity (assuming no air resistance).

The equation of motion that relates distance travelled, time, initial velocity and acceleration is a quadratic:

s = ut + 0.5gt^{2}

Where:

- s is the distance travelled at time t.
- u is the initial velocity.
- g is the acceleration.

Note that in the projectile example, the height and the initial velocity are both positive (ie upwards), but the acceleration due to gravity acts in the opposite direction and so has a negative value. So the coefficient for the squared term will be negative, giving the inverted parabola shown in the second example in Section 2.

If we put some values into the equation, g = -10 m s^{-2} and u = 50 m s-1, we can then plot the height of the projectile over time:

Note that this plot has a horizontal axis of time, but it looks very much like the path that a projectile would take in space if it were fired at an angle to the horizontal rather than vertically. We can use the quadratic equation to construct a model of projectile flight. The first part is to work out how fast the projectile is travelling vertically and how fast it travels horizontally when fired from a gun at an angle θ to the horizontal.

- Initial vertical velocity is given by:
- u
_{v}= u sinθ

- Horizontal velocity is assumed to be constant:
- u
_{h}= u cosθ

- So at a given time t, the height of the projectile is:
- s
_{v}= u_{v}t + 0.5gt^{2}

- and the distance travelled horizontally is:
- s
_{h}= u_{h}t

We can calculate a trajectory for any values of θ between 0° (horizontal) and 90° (vertical). Taking a muzzle velocity u = 50 m s^{-1}, and muzzle elevation angles θ = 30° and θ = 45° gives the following trajectories:

*The muzzle elevation of θ = 45° gives a greater vertical component of velocity, so that the projectile not only travels to a greater height, but also travels further horizontally.*