A Bouncing Basketball

Seeing a banked three-point shot from each of my alma maters last night got me thinking about bouncing basketballs.  Hold a basketball at a certain height and drop it onto the floor.  You will notice that it does not return to the same height from which it was dropped.  The ball in fact returns to a lower height because of energy lost during the collision with the floor.  There is a little energy lost to air resistance as the ball moves through air, but that loss is small compared to the energy lost during the collision.  It is actually easy to tell that energy is lost, even with your eyes closed!  The fact that you hear the ball bounce off the floor means that your ear picks up a sound wave that carries energy.  Energy is also lost to heat as the basketball's surface rubs against the floor.  The reason the ball rubs against the floor is because the ball deforms during the collision, meaning its surface spreads slightly against the floor.  Using more technical language, the ball scrunches on the floor.


A parameter we use to tell us something about energy loss is the coefficient of restitution or COR, which is the ratio of the speed just after the bounce to the speed just before the bounce (this assumes the ground does not move).  Because those speeds are sometimes hard to measure, we use heights.  A simple way to calculate COR for the dropped basketball experiment that I mentioned in the previous paragraph is COR =( h_f / h_i )^1/2, where h_i is the initial height off the floor and h_f is the rebound height after the bounce.  According the NCAA rules (click here for the rule book and go to page 25), a ball dropped from a height of 6 feet (1.83 m) must rebound to a height of "not less than 49 inches" (1.24 m) and "not more than 54 inches" (1.37 m) for a college men's ball.  A college women's ball must rebound between 51 inches (1.30 m) and 56 inches (1.42 m).  That means that for a college men's ball, 0.825 < COR < 0.866, and for a college women's ball, 0.843 < COR < 0.882.  A referee can check a ball before a game by dropping it from the height of his or her head and seeing if the ball bounces to a height roughly 2/3 to 3/4 of the referee's height, which is roughly the height of the referee's mid torso.  A ball that does not satisfy the referee will need to have its air pressure changed if there is nothing wrong with the ball itself.


The amount of energy the ball keeps after the collision scales with the square of COR.  A college men's ball thus retains between 68% and 75% of its energy after colliding with the floor.  For the college women's ball, between 71% and 78% of the ball's energy is retained.  A guard dribbling the ball down the floor puts energy into the ball by doing work on it, namely by exerting a downward force on the ball while displacing the ball downward.  One must keep pushing the ball downward if one wants it to return to the same height each time because a ball with an initial downward velocity from a height of a player's hip can return to the player's hip height, whereas a ball dropped (i.e. zero initial velocity) from the player's hip height cannot.


Note that COR depends on the properties of the two surfaces involved in the collision.  One cannot simply quote a COR for a given sports ball without also specifying the surface the ball hits.  For the COR calculations I did here, I used the specifications in the rule book, which describe the ball colliding with the "playing surface."  For a ball bouncing off glass, like in the banked three-point shots I saw last night, COR is likely to be different from the value measured by dropping a ball on the playing surface.


A bank shot is even more interesting than a ball dropped onto the floor because the ball banking off the glass not only feels a force from the glass that is perpendicular to the glass surface, the ball feels a frictional force parallel to the glass surface.  That frictional force creates a torque on the ball, which changes the ball's rotation.  That means that the rebound angle the ball makes with the glass is not the same as the angle the ball made with the glass just before the ball hit the backboard.  Basketball players know this -- just watch where they bank the ball off the glass on layups.


As always, enjoy the wonderment of a sporting moment first, and then think about physics later.  After both my schools won last night, I was too happy to write about physics!