# Why does height affect bounce efficiency?

I recently conducted an experiment, where I dropped tennis balls from various heights and recorded their rebound heights. The higher the drop, the higher the rebound height, obviously. However, the higher the drop, the smaller the bounce efficiency also became. For example, a drop from 50cm had a 38 cm rebound whereas a drop from 200cm had a 102cm rebound. That is, the ratio becomes smaller with higher drops.

Why is this so?

Is it something to do with air resistance? Theoretically, shouldn't the ball have the same bounce efficiency no matter the height?

When a ball bounces, it is deformed at the contact point to become a compressed spring, having almost all of the kinetic energy converted to spring compression, as the velocity downward goes to zero. It then is forced upward by the springiness of the material of the ball.

The efficiency of the bounce depends on how greatly the ball is deformed; a thick rubber ball may lose more energy to internal heating if it is deformed to a near-hemisphere, though be very lively when a low-velocity impact makes a small dent.

Also, some ball structures may include viscous or viscoelastic materials (which have different mechanical constants at different strain rates). A fast impact would imply a high strain rate, but the rebound may be slow by comparison. Such materials don't make good toys, but are excellent for sound-deadening.

One way of determining the hardness of steel is to bounce a hardened ball against a sample: the hardest steel does the least energy-absorbing deformation, so returns the ball higher than softer metal.

In the case of a tennis ball, remember that a long drop creates more airspeed-related losses than a short one; there's more than rebound energy efficiency involved in that experimental setup.

Is it something to do with air resistance?

There's a small effect for these smallish velocities, but that's probably not what you're seeing. The terminal velocity of a tennis ball is about 30 m/s. At the bottom of your 200 cm drop, the ball is going a bit over 1/5 of terminal velocity. Since drag is proportional to the velocity squared, it's a small effect. If you were dropping the ball from 100 meters vs a kilometer, drag would be a big effect. The ball would bounce about the same height for those large drop heights.

Theoretically, shouldn't the ball have the same bounce efficiency no matter the height?

In a spherical cow theory (the canonical example of an overly simplified physics model), yes. In practice, no. The coefficient of restitution is velocity dependent. Drag is but one factor for why this is the case. The ball undergoes some rather significant and nonlinear distortions during the bounce.

You measured a bounce of 102 cm for a 200 cm drop. That means your ball isn't legal. A legal tennis ball has to have a 138 to 151 cm rebound from a hard surface when dropped from a height of 254 cm under carefully controlled lab conditions. Your ball is a bit too lossy, so it's possible that what you measured is correct (for that ball).

There are many ways in which a physics experiment can go awry. Did you make multiple measurements? Did you use the same conditions (e.g., same ball, same temperature of the ball and of the air, same relative humidity) for each drop test? Does your release mechanism introduce a bias? Does your rebound measurement introduce a bias? Addressing the issues that can cause an experiment to go awry is very important with physics experiments.

The ball dropped for a low height like 20 cm will give a lower rebound height than the higher drop heights until it reaches its terminal velocity when it is the fastest downward force it will have due to air resistance. (not throwing, dropping) Therefore the bounce efficiency will decrease and decrease.

Just my guess - The ball deforms when it hits the ground. As you drop from higher and higher heights, it deforms more and more. But there's a limit to how much it can deform and as you approach this limit the amount of deformation achieved increases more slowly with height. So as you drop from greater heights, I wouldn't expect the rebound to be proportional to the height from which the ball is dropped.

Also based on the surface at the point of contact of the ball. Surface friction, deformation and viscosity of the surface(glass/wood/concrete/rubber)