A ball is dropped from a height I on the ground. If the coefficient of restitution is e, the height to which the ball goes up after it rebounds for the nth time is

Let v o be the velocity with which the ball strikes the earth first time and v n be the velocity after nth rebound. According to the definition of coefficient of restitution, we have e=V1V0=V2V1=V3V2=…=VnVn−1 Further en=v1v0×v2v1×v3v2×…×vnvn−1=vnv0 Here v0=2gh  and  vn=(2gH) ∴ en=2gH2gh=Hh  or  e2n=Hh  or  H=he2n