Projectiles from a Height

The question

From a clifftop, is 45° still the best angle?

No. From a clifftop the best angle is below 45°: at 15 m/s it is about 31° from a 20 m cliff and about 25° from a 40 m cliff. Below the edge the ball gets extra falling time whatever the angle, and that time is worth more when more of the velocity is horizontal. So the higher the cliff, the flatter the best throw.

How to solve a projectile launched from a height

1. Split the launch velocity: $v_x=v\cos \alpha$, $v_y=v\sin \alpha$.
2. Take up as positive, so the landing is at a negative displacement, $s=-h$, and $a=-9.8$ m/s².
3. Solve $s=v_{y}t+\frac{1}{2}a{t}^2$ for the time. It is a quadratic; take the positive solution.
4. Range = $v_x$ × that time.