A ball is released from height h w.r.t. the ground. Draw a qualitative height versus time diagram of the ball bouncing in a non-ideal case.

In a non-ideal case, there will be energy loss in heat when the ball touches the ground. In particular:K_n=aK_n-1, where K_n-1 is the kinetic energy before the nth bounce, K_n is the kinetic energy after the bounce, and a is the fraction of kinetic energy that remains after the bounce. We can see that this produces a geometric series of the form: K_n=aⁿK₀, which gives the kinetic energy after n bounces. To transform this into height, we just need to remember that the maximum height after n bounces h_n is reached when K_n is all converted into potential energy (mgh), where m is the mass of the ball. Substitute the formula of KE. Hence: h_n=K_n/(mg)=aⁿK₀/(mg). Now, as K₀ is proportional to the height to which the ball was originally released h₀(again, for the conservation of energy), We get: h_n \propto aⁿh₀. Hence the maximum height decrease exponentially (as a <1). The maximum vertices are also peaks of rotated parabolas, as the ball obeys the free-fall equation which says that h is proportional to t². Draw this and you get the diagram requested.