a). A ball is thrown vertically upwards with a speed of 20 metres per second from a point 3 metres above the ground. Find the speed with which it hits the ground.
b). If the ball rebounds with a speed two thirds of the speed it hits the floor, find the greatest height it reaches after bouncing.
c). After how many bounces is the ball subsequently always less than 5cm from the ground?
I have done part a and part b however I have no idea how to do part c.
Let $V_n$ denote the speed of the ball after the $n$th time the ball hits the ground. You know the speed at which it hit the ground, let us call it $\alpha$, then: $V_1 = \frac23 \alpha$.
$(V_n)_n$ is a geometric sequence, and we find: $V_n = (\frac23)^n \alpha$, for each $n \ge 1$.
Find a formula $H_n$ for the height that the ball reaches after the $n$th bounce, and solve $H_n \le 0.05$