A ball of mass m travelling at a speed v undergoes a head-on collision with an identical ball at rest. Following the collision, the total kinetic energy of the balls is three-fourths of its initial value. Determine the coefficient of restitution.

Step-by-step solution

The loss in kinetic energy during a head-on collision is given by: K = 1 2 m₁ m₂ m₁ + m₂ (u₁ - u₂)^2 (1 - e^2) Here, m₁ = m₂ = m , u₁ = v , and u₂ = 0 . K = 1 2 m m m + m v^2 (1 - e^2) = 1 4 m v^2 (1 - e^2) Given that the final kinetic energy is three-fourths of the initial kinetic energy, the loss in kinetic energy is one-fourth of the initial kinetic energy: K = 1 4 K_i = 1 4 ( 1 2 m v^2 ) = 1 8 m v^2 Equating the two expressions for K : 1 4 m v^2 (1 - e^2) = 1 8 m v^2 1 - e^2 = 1 2 e^2 = 1 2 e = 1 2