## Thursday, August 10, 2017

### That's the way the ball bounces.

How does a ball bounce?  Why does a ball, dropped from some height onto a flat surface, not bounce all the way back up to its starting height?  The answers to these questions may seem obvious, but earlier this week, this paper appeared on the arxiv, and it does a great job of showing what we still don't understand about this everyday physics that is directly relevant for a huge number of sports.

The paper talks specifically about hollow or inflated balls.  When a ball is instantaneously at rest, mid-bounce, it's shape has been deformed by its interaction with the flat surface.  The kinetic energy of its motion has been converted into potential energy, tied up in a combination of the elastic deformation of the skin or shell of the ball and the compression of the gas inside the ball.  (One surprising thing I learned from that paper is that high speed photography shows that the non-impacting parts of such inflated balls tend to remain spherical, even as part of the ball deforms flat against the surface.)  That gas compression is quick enough that heat transfer between the gas and the ball is probably negligible.  A real ball does not bounce back to its full height; equivalently, the ratio $v_{f}/v_{i}$ of the ball's speed immediately after the bounce, $v_{f}$, to that immediately before the bounce, $v_{i}$, is less than one.  That ratio is called the coefficient of restitution.

Somehow in the bounce process some energy must've been lost from the macroscopic motion of the ball, and since we know energy is conserved, that energy must eventually show up as disorganized, microscopic energy of jiggling atoms that we colloquially call heat.   How can this happen?

• The skin of the ball might not be perfectly elastic - there could be some "viscous losses" or "internal friction" as the skin deforms.
• As the ball impacts the surface, it can launch sound waves into the surface that eventually dissipate.
• Similarly, the skin of the ball itself can start vibrating in a complicated way, eventually damping out to disorganized jiggling of the atoms.
• As the ball's skin hits the ground and deforms, it squeezes air out from beneath the ball; the speed of that air can actually exceed the speed of sound in the surrounding medium (!), creating a shock wave that dissipates by heating the air, as well as ordinary sound vibrations.  (It turns out that clapping your hands can also create shock waves!  See here and here.)
• There can also be irreversible acoustic process in the gas inside the ball that heat the gas in there.
This paper goes through all of these, estimates how big those effects are, and concludes that, for many common balls (e.g., basketballs), we actually don't understand the relative importance of these different contributions.  The authors propose some experiments to figure out what's going on.  The whole thing is a nice exercise in mechanics and elasticity, and it's always fun to realize that there may still be some surprises lurking in the physics of the everyday.

Speaking of balls bouncing, I wish physicists knew of the Hertz law (collisions of spheres has $F \prop x^{3/2}$, a wall is just a sphere of infinite radius) so they could think a bit more beyond simple spring-mass approaches. Mechanical impacts is a very rich field of study.