A ball impinges directly on a similar ball at rest. The first ball is brought to rest by the impact. If half of the kinetic energy is lost by impact, the value of coefficient of restitution is?

Let u1 and v1 be the initial and final velocities of ball 1 and u2 and v2 be the similar quantities for ball 2. Here, u2 = 0 and v1 = 0.
∴ initial KE, Ki = 1/2 mu1² + 1/2 mu2² = 1/2 mu1²
and final KE, Kf = 1/2 mv1² + 1/2 mv2² = 1/2 mv2²
Loss of KE, ΔK = Ki - Kf = 1/2 mu1² - 1/2 mv2²
According to the question, 1/2(1/2 mu1²) = 1/2 mu1² - 1/2 mv2² (∵ half of its KE is lost by impact)
or u1² = 2v2²
or v2 = u1/√2
∴ Coefficient of restitution, e = |v2 - v1| / |u1 - u2| = v2 / u1 = 1/√2.