A small ball is released from rest from a point that is 40m above horizontal ground. The ball bounces on the ground and rebounds vertically. Each time the ball bounces on the ground, the speed of the ball is instantaneously reduced by 50%. The ball is modelled as a particle moving freely under gravity, from the instant when it is released until it first hits the ground, and between each successive bounce. (a) Find the time from the instant when the ball is released from rest to the instant when it hits the ground for the second time.
(b) Find the maximum height reached by the ball above the ground after the ball's third bounce. ​