You are standing on a cliff at a height h above the sea. You are capable of throwing a stone with velocity v at any angle a between horizontal and vertical. What is the value of a when the horizontal distance travelled d is at a maximum?
On level ground, when h is zero, it's easy to show that a needs to be midway between horizontal and vertical, and thus π/4 or 45°. As h increases, however, we can see by heuristic reasoning that a decreases to zero, because you can put more of the velocity into the horizontal component as the height of the cliff begins to make up for the loss in the vertical component. For small negative values of h (throwing up onto a platform), a will actually be greater than 45°.
Is there a fully-solved, closed-form expression for the value of a when h is not zero?