A ball is dropped from a height of 2m onto a perfectly vertical surface. After each impact with the surface, the ball rebounds to 60% of the height from which it just fell. Neglect air resistance and assume the time spent in contact with the surface is negligible.
a. Part 1: (a) Let Dn be the total vertical distance the ball has traveled by the time it reaches its maximum height after the nth rebound. Write an expression for Dn in terms of n.
b. Part 2: (b) Determine the exact limiting value of Dn as n→∞, and state this value in meters.
c. Part 3: (c) Find the least integer n such that Dn>9m. Show all reasoning clearly.