Vertical bounce of two vertically aligned balls
(a) Two masses headed toward the end stop of an air track at the same speed . (b) Mass reflects at speed and collides with . If , continues to the right at reduced speed as in (c); otherwise it comes to a stop or heads back to the end stop as in (d). The subsequent behavior is shown in (e) and (f).
(a) The ratios and versus for or 1.0. If , reflects off the end stop a second time at speed . (b) The ratios and versus for or 1.0. These solutions apply when . If , reflects off the end stop a third time at speed .
If remains in contact with the end stop and with during the collision, the collision can be analyzed using connecting springs and . The same situation arises when one ball sits on another and both fall to the floor together. and denote the displacements of the two masses.
Solutions of Eqs. (5) and (6) when (a) and (b) .
Solutions of Eqs. (5) and (6) with , , , and . During the interval , travels back toward the end stop at constant speed, and travels away from the end stop at constant speed.
Rebound speed ratios (dashed curves) and (solid curves) versus the stiffness ratio for three values of the mass ratio .
Force versus compression measured under static conditions for each ball separately and with one on top of the other as shown in the inset.
Force waveforms measured when a tennis ball is on top of a basketball and when both are dropped to the floor. The piezoelectric P1 on the floor generates a voltage proportional to force on the bottom of the basketball. The P2 signal is proportional to the force between the two balls.
A solution of Eqs. 5 and 6 approximating the experimental results in Fig. 8.
(a) Force waveforms measured with ball B2 on top of B1 and when both were dropped together. P1 generates a voltage proportional to the force on the bottom of the bottom ball. The P2 signal is proportional to the force between the two balls. (b) Result when B1 is on top of B2.
(a) Force waveforms measured when the small superball B3 is on top of B1 and when both are dropped together. The P2 response from 0 to and at was also present when B1 was dropped on its own and is due to vibration of the upper surface of B1. (b) Result when B1 is on top of B3.
(a) Force waveforms measured using two large superballs. (b) Result when P2 is replaced by two copper strips to measure the contact resistance between the two balls.
Mass-spring calculation for the B1-B2 combination assuming that the balls were spaced apart as they fell and that the coefficient of restitution of the B1 ball was 0.78. The experimental result is shown in Fig. 10(a) (where P1 is proportional to , and P2 is proportional to ).
Superball properties. is the ball diameter, is the ball mass, and COR is the coefficient of restitution.
Values of and , for each ball pair. Bottom (top) refers to the bottom (top) ball. The last entry is the result for the tennis ball on top of the basketball.
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