A projectile is fired in such a way that its horizontal range is equal to 7 times its maximum height. What is the angle of projection?

The projectile is fired with speed at an angle relative to the horizontal, so that the horizontal components of the initial velocity vector are

Its position vector has components

where

Recall that

where and are the projectile's initial and final velocities, is its acceleration, and is its displacement.

In the vertical direction, velocity is 0 at maximum height and , so that

The projectile hits the ground when , which happens for

where we omit because we know the projectile starts on the ground. This means the maximum horizontal range is

We're given that , so

Solve for :

The first case suggests that , but then both the maximum height and range would be 0, which would technically satisfy the given condition, but it's not an interesting solution.

In the second case, we get

So the projectile was launched at an angle of about 29.7º.