The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
Bouncing Balls and Geometric Series, Robert Styer and Morgan Besson
During the actual bounce ("the collision"), the super ball loses some energy and some momentum. How is this related to the maximum height, the velocity of the ball, and the time? The basic connections are:
Maximum Height ↔ Energy
Bounce speed ↔ Momentum ↔ Time for Bounce
We claim that if each bounce rises a fraction of the previous height (e.g. 80%), then the energy of the ball between bounces changes by the same fraction (e.g., 80%), and the time, speed and the linear momentum change by the square root of this fraction (e.g., square root of 80%, about 90%).
Consider the energy. The total energy is constant between bounces if we neglect air friction. We have the kinetic energy of the ball and the potential energy due to gravity. At the top of the bounce, there is no kinetic energy so all the energy is potential.
Standard physics textbooks show that the potential energy of the ball at height h is given by a formula like mgh, where m is the mass of the ball and g is the familiar gravitational acceleration constant 9.8 m / s2 or 32 feet / s2. If each bounce rises a fraction R of the previous height, the energy changes from mgh to mg(Rh) = R(mgh) so the energy changes by the same fraction R.
We claim that if each bounce rises a fraction R of the previous height, then the speed of the ball just after each bounce changes by a factor of the square root √R. Standard textbooks on kinematics show that if an object is launched straight up with speed v, then the maximum height reached will be h = v2 / (2g) so v = √(2gh). We see that changing the bounce height from h to Rh will change the bounce speed from v to (√R) v. Note that the same square root factor appears in the time change of each bounce as explained in the discussion of bounce height and time. Since linear momentum is mv, linear momentum changes by the same r = √R factor at each bounce.
For example, if a ball bounces 80% of its height on each bounce, then the ball is losing 20% of its energy on each bounce. The time of each bounce is about 90% of the time of the previous bounce, the ball slows down about 10% each bounce, and about 10% of the linear momentum is lost at each bounce.
Thus, the elasticity coefficient, R, is the ratio of the bounce heights, that is, the ratio of the energy of successive bounces. The coefficient of restitution, r = √R, is the ratio of the velocities, equivalently, the ratio of successive momentums or successive bounce times.