A sphere of mass 'm', moving with velocity '3u' collides head-on with another identical sphere at rest. If ' e ' is coefficient of restitution then what will be the ratio of velocity of the second sphere to that of first sphere after collision?

Step-by-step solution

Let m be the mass of each sphere, with initial velocities u₁ = 3u and u₂ = 0 . After collision, their velocities are v₁ and v₂ . Conservation of momentum gives m(3u) + m(0) = mv₁ + mv₂ , so v₁ + v₂ = 3u . The coefficient of restitution provides e = v₂ - v₁ 3u , yielding v₂ - v₁ = 3eu . Adding these equations eliminates v₁ : 2v₂ = 3u(1 + e) , so v₂ = 3u(1 + e) 2 . Subtracting gives 2v₁ = 3u(1 - e) , so v₁ = 3u(1 - e) 2 . The required ratio is v₂ v₁ = 1 + e 1 - e , matching option B .