A projectile is launched from a horizontal floor with speed 20 m/s at 45°. Every collision with the smooth floor has coefficient of restitution e = 1/2, and there is no horizontal impulse during collision. Take g = 10 m/s². If the total horizontal distance travelled before the infinite sequence of bounces ends is N m, find N.
Physics · Center of maas , Momentum and Collision · collision and types of collision
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Question

A projectile is launched from a horizontal floor with speed 20 m/s at 45°. Every collision with the smooth floor has coefficient of restitution e = 1/2, and there is no horizontal impulse during collision. Take g = 10 m/s². If the total horizontal distance travelled before the infinite sequence of bounces ends is N m, find N.

Physics · Center of maas , Momentum and Collision · collision and types of collision

G1 Physics · Accepted Answer

The initial projectile range before the first impact is R1 = u²sin2θ/g = 20²×1/10 = 40 m. After every floor collision, the vertical component of velocity is multiplied by e, while the horizontal component remains unchanged. Therefore each subsequent flight time, and hence each subsequent horizontal range, is multiplied by e. Thus the total horizontal distance is the geometric series: R(total) = R1[1 + e + e² + e³ + ...] = R1/(1−e). With R1 = 40 m and e = 1/2: R(total) = 40/(1−1/2) = 80 m. Hence…