Introduction to Classical Mechanics · Problem 3.40

Using F = ma — Morin

Physics — Mechanics · redirected_motion

Problem

Redirected motion * A ball is dropped from rest at height hh. At height yy, it bounces off a surface with no loss in speed. The surface is inclined at 45∘45^\circ, so the ball bounces off horizontally. What should yy be so that the ball travels the maximum horizontal distance? What is this maximum distance?
Intuitive explanation

The key is recognizing that the total horizontal distance depends on two competing effects: the ball gains more speed if it bounces lower (larger fall distance), but it has less time to travel horizontally afterward. You need to find the sweet spot where the product of horizontal velocity and remaining flight time is maximized. Set up the speed at bounce height yy using energy conservation, express the horizontal distance traveled after the bounce, then optimize by taking a derivative with respect to yy.

Intuitive explanation

The key is recognizing that the total horizontal distance depends on two competing effects: the ball gains more speed if it bounces lower (larger fall distance), but it has less time to travel horizontally afterward. You need to find the sweet spot where the product of horizontal velocity and remaining flight time is maximized. Set up the speed at bounce height yy using energy conservation, express the horizontal distance traveled after the bounce, then optimize by taking a derivative with respect to yy.

Browse the problem library