A projectile is launched from the top of a cliff of height h with fixed speed u. Its direction of projection can be chosen freely.
What is the maximum possible horizontal range measured from the vertical through the launch point to the point where it hits the horizontal ground?

Question

A projectile is launched from the top of a cliff of height h with fixed speed u. Its direction of projection can be chosen freely.

What is the maximum possible horizontal range measured from the vertical through the launch point to the point where it hits the horizontal ground?

G1 Physics · Accepted Answer

Let the impact velocity be v⃗. By energy conservation, v² = u² + 2gh. The horizontal range is R = u cosθ·T. For the optimum trajectory, the standard maximum-range-from-height result is R m a x R_max R m ​ a x = (u/g)√(u²+2gh). It can also be obtained by maximizing R(θ) = (u cosθ/g) [u sinθ + √(u²sin²θ+2gh)] with respect to θ. Therefore, R m a x R_max R m ​ a x = (u/g)√(u²+2gh).