A startling version of the two-ball demonstration was recently published by Joseph L. Spradley of Wheaton College in Wheaton, Ill. Hold a baseball above a basketball (it is best to maintain a slight separation) and drop the pair from waist height. The basketball goes almost dead on the floor; the baseball reaches the ceiling. Here safety precautions are a must. I once was bruised when I failed to align the balls properly, so that the baseball shot out sideways like a cannonball, moving faster than I could.

If you drop a lightweight plastic Wiffle ball instead of the baseball, the basketball bounces a little higher and the Wiffle ball actually slams into the ceiling. Even more spectacular launches can be made with a stack of three balls, provided the mass of the balls decreases toward the top of the stack. If all went ideally, the top ball would be propelled to a height that is 49 times its release height. Practical matters reduce the height, but it can still be dramatic.

When a falling stack of balls reaches the floor, the balls undergo a chain of collisions in which kinetic energy is transferred upward through the stack. The last collision reverses the top ball's motion and increases its speed. The height to which it then climbs depends on the square of the speed at which it moves just after the collision. If the stack has only two balls, the collision can at best triple the top ball's speed, sending it up to nine times the release height. With three balls in a stack, the collision can at best increase the top ball's speed sevenfold, so that it goes 49 times higher than its release height. You might think the greatest height would be achieved if all of the bottom ball's energy were transferred to the top ball, but that is not the case. As will be seen below, the highest bounces come when only a fraction of the bottom ball's energy is given to the top ball.

Exactly how is the energy transferred between the balls? As a warmup for the explanation I shall consider simpler collisions. For a start, picture one ball moving to the right and striking a second, initially stationary ball. Assume that the balls are isolated so that we need not worry about any extra forces on them, such as friction from the surface over which they move. Also assume that the balls collide head-on. During the collision some or all of the first ball's kinetic energy is transferred to the second ball, and the second ball moves to the right. Depending on the details of the collision, the first ball may move to the right or to the left or may even be stationary. The questions are: In a given situation how much energy is transferred, and what speed is imparted to the second ball?

In 1968 John B. Hart and Robert B. Herrmann, who were then at Xavier University in Cincinnati, reported theoretical studies of such collisions. They also conducted experiments with steel balls of various sizes suspended by threads from a rod. To study how energy is transferred in a collision, several balls were hung in a row, with slight separations between them, as in the illustration at the right. The idea was to draw back a ball at one end of the row, release it and then see how the subsequent collisions transferred energy to the ball at the other end. The initial energy of the first ball was set by the height from which it was released. The energy finally imparted to the last ball could be measured by the height to which it swung after being hit. In this arrangement, incidentally, the balls were about as isolated as one could hope for.

Here I shall review only the theoretical findings by Hart and Herrmann, but you can check the results, as they did, with their pendulum apparatus. Consider the case I described above in which a ball strikes another, stationary ball. Two factors are important: momentum and kinetic energy. Momentum is the product of mass and velocity; kinetic energy is half the product of the mass and the square of the speed. Since the balls are isolated, the momentum of the balls can be exchanged in a collision, but the total momentum must remain the same. Such a firm rule does not usually apply to the kinetic energy, which is reduced when some of it is transformed into sound or goes into vibrations or deformations of the balls. If there were no such losses of kinetic energy, the collision would be perfectly elastic. Everyday collisions, however, are not so ideal, and they are said to be inelastic. For example, you can nearly always hear a collision, and so some energy must go into sound.

One way to symbolize the extent of elasticity is to employ a parameter called the coefficient of restitution. A perfectly elastic collision has a coefficient of exactly 1, whereas a completely inelastic collision has a coefficient of zero. A collision of steel balls, for example, may have a coefficient as high as .99; a collision between a baseball and a basketball has a smaller coefficient. Although a higher coefficient means that more energy is delivered to the second ball, this is no guarantee that the ball will in fact gain a lot of energy. Even if the total kinetic energy remains essentially constant in the collision, the first ball may transfer energy to the second ball only grudgingly.

The extent to which energy is transferred depends not only on elasticity but also on the ratio of the first ball's mass to that of the second ball. Let's go back to the case in which one ball runs into a stationary second ball. If the mass ratio is one and the collision is perfectly elastic, the full energy of the first ball is given to the second ball, and the first ball stops [see illustration at right above]. For any other ratio of masses, either smaller or larger, the transfer is less. For example, if the first ball is 100 times as massive as the second one, only about 4 percent of the energy is transferred. Surprisingly, the equation that predicts this transfer has a symmetry. For a given mass ratio it does not matter which ball is initially moving: the same fraction of energy is transferred if the roles are reversed and the lighter ball runs into a stationary, heavier ball. In both cases the transfer is small because the masses are mismatched. The larger the mismatch, the poorer the transfer.

If a third ball of intermediate mass is inserted between the mismatched balls, the transfer improves. Now there is a chain of two collisions, and in each one the colliding masses are more closely matched than they were in the original single collision. Hart and Herrmann found that the transfer is best when the intermediate ball has a mass equal to the geometric mean of the masses of the other balls. (The geometric mean is the square root of the product of the masses.) In my example, the intermediate ball should have a mass 10 times the lighter ball's mass. When the third ball is put in place, the energy transfer jumps to about 11 percent. The equation associated with the transfer still has symmetry: it does not matter whether the chain of collisions begins with the heaviest ball or the lightest one. Aside from the extent of elasticity in the collisions, only the mass ratios affect the energy transfer.

The transfer may improve if even more balls of intermediate mass are inserted into the chain. Hart and Herrmann found that the transfer is optimum if the mass ratios of successive balls are identical. This condition is the same as requiring that each intermediate ball have a mass that is equal to the geometric mean of the masses of the balls next to it.

For example, if the ratio of the first ball's mass to the second ball's mass is 1.05, then the ratio of the second ball's mass to the third ball's mass should also be 1.05, and so on. That is about the right ratio if there are 100 balls inserted between the original two balls, which have a mass ratio of 100. The first, heaviest, ball strikes a ball that is only slightly less massive, and the first ball gives up nearly all of its energy. The second ball then collides with the third one, which is only slightly less massive, and again the transfer is nearly perfect. When the last ball is struck, it receives almost 95 percent of the energy the first ball had initially. Symmetry still holds: the same percent of energy is sent down the line of balls if the transfer goes in the opposite direction.

If even more balls are inserted in the line and the masses are adjusted so that the mass ratio between successive balls is identical throughout the line, the transfer nears 100 percent. The reason is that with a longer chain the balls in each colliding pair are closer to being exactly matched in mass (in which case all of the energy would be transferred). If the balls are hung as adjacent pendulums, the release of the ball at one end sends energy through the chain until the last ball finally swings out and up. During the successive collisions the intermediate balls hardly move and there is little evidence of the energy transfer except for the clatter that sweeps along the line of balls.

So far, the collisions have been considered to be perfectly elastic. The fact that in practice most collisions are inelastic changes the story. When the coefficient of restitution is less than the ideal value of 1, the transfer of energy worsens if the chain of balls is too long. Although a long chain means that the balls in each colliding pair are nearly matched in mass, the steady drain of energy from the balls into sound, vibration and deformation diminishes the energy reaching the last ball.

Hart and Herrmann calculated just how many intermediate balls are needed to maximize the energy transfer to the last ball for a given coefficient of restitution and for a given mass ratio between the end balls. If there are fewer balls in line, the consequent larger mismatch in mass between the members of each pair decreases the transfer; if there are more balls, the inelasticity of the collisions lowers the transfer. For example, suppose that the mass ratio of the end balls is again 100 but that now the coefficient of restitution is .99-only slightly less than perfect. The maximum transfer takes place when there are 22 intermediate balls. If the coefficient is .8 there should be only four intermediate balls. And if the coefficient is as low as .19, the best transfer (less than 2 percent) is obtained when the first ball hits the last ball directly. In all the examples, the transfer maintains symmetry.

What conditions will maximize the speed imparted to the last ball? Start again with the case in which there are only two balls and the collision is perfectly elastic. More speed is imparted to the second ball if its mass is small compared to that of the first ball. In the limit where the mass ratio is infinite, the second ball is given a speed that is twice the first ball's initial speed. With such a mass ratio, however, the energy transfer is minuscule. The result may be perplexing. How can the second ball be given its greatest speed when it gains only a tiny amount of energy? The answer lies in the fact that its mass is so small. If it is given even a small amount of energy, its speed will be large.

Here is one way to derive the speed of the second ball without resort to any equations. Let V be the speed of the first ball. When the mass ratio is very large, the speed of the first ball hardly changes during the collision. Picture the collision from the perspective of the first ball, as if you could somehow ride along with it [see illustration below]. Before the collision the second ball (which is actually stationary) appears to approach you and the first ball with a speed V. If the collision is perfectly elastic, the second ball appears to bounce off the first ball and then move backward with a speed of V relative to the first ball. Since the first ball still has a speed that is approximately V, the speed of the second ball is actually V + V, or 2V. In 1972 James D. Kerwin of the California State Polytechnic University in Pomona reported calculations on a chain of collisions, where each collision is perfectly elastic and involves a massive ball hitting an infinitely lighter ball. The speed doubles with each collision and, if there are n intermediate balls, the last ball ends up with a speed that is 2n times the first ball's speed. Obviously a long chain results in a fantastic final speed.

Several lessons can be learned from these examples of chain collisions. The extent of energy transfer depends on elasticity and mass ratios; if the masses are chosen properly, the last and lightest object can end up with much of the energy or with a large speed, but the mass ratios required for those two end results differ.

These lessons apply to the chain collisions when a stack of balls is dropped on the floor. The ball on the bottom rebounds from the floor and then runs into the second ball in the stack. The second ball rebounds from the first ball and then runs into the third ball, and so on until the top ball is reached. In each collision in the chain, the energy transfer and the speed imparted to the higher ball depend on elasticity and mass ratios.

The high rebound of a ball from a dropped stack of balls was first reported by Walter Roy Mellen in 1968, not long after the introduction of the Super Ball by the Wham-O Manufacturing Company. (A Super Ball is considerably more elastic than a common rubber ball.) Mellen described putting a small Super Ball on top of a larger Super Ball and dropping the pair. To keep them aligned during the fall, he sometimes stuck a drop of glue or a strip of double-sided tape between them. (He said that neither technique noticeably altered the high rebound of the smaller ball, but my experience is that the effect is more pronounced if there is a slight separation between the balls.) He obtained even larger rebounds when a table-tennis ball was positioned above the smaller Super Ball and the stack of three balls was dropped. (Although a table-tennis ball is larger than the smaller Super Ball, it is lighter, and it is the mass ratio that counts.) Typically the table-tennis ball would shoot up to about 20 times the release height.

Gerhard Stroink of Dalhousie University and several other authors have suggested an easy way to picture what happens to the balls when they are dropped. Start with two balls, the upper one of which has a much smaller mass than the lower one. Assume that the collisions between ball and floor and between ball and ball are perfectly elastic. Let V represent the speed of the balls before the lower ball hits the floor. Just after the lower ball bounces, it heads upward with speed V toward the top ball, which is still headed downward with speed V [see illustration at right]. The balls close on each other at a rate that is the sum of their speeds, pr 2V.

Imagine the impending collision from the perspective of the lower ball. The second ball approaches with a speed of 2V, bounces off the lower ball and then heads upward at a speed of 2Vwith respect to the lower ball. Since the mass ratio is large, the lower ball is still moving upward at a speed of almost V with respect to the floor. Hence the top ball must have a speed of V + 2V, or 3V, relative to the floor. Recall that the height to which a ball bounces depends on the square of its speed right after the collision. In this case, where the second ball's speed is tripled by the collision, it bounces to nine times its release height.

Now add a third, even lighter ball to the top of the stack and imagine the second and third balls just before they collide. The second ball is headed upward at a speed of 3Vand the third ball is headed downward at a speed of V. The balls close on each other with a relative speed of V + 3V, or 4V. After the collision, the third ball heads upward with a speed of 4V relative to the second ball. Since the second ball has a speed of 3Vrelative to the floor, the third ball must have a speed of 3V + 4V, or 7V, relative to the floor. In the ideal setting of infinite mass ratios and perfectly elastic collisions, the third ball should rise to a height 49 times its release height. You may want to continue the analysis to stacks of four or more balls.

Let's return now to the case in which only two balls are dropped and again assume that the collisions are perfectly elastic. If you want a full transfer of energy between the balls so that the lower ball stops when they collide, you must arrange for the mass ratio to be exactly three, in which case the top ball reaches four times its release height. The rebound height is not as dramatic as when the mass ratio is much larger, but the demonstration is still surprising. Here are two balls that bounce well when dropped separately, and yet when they are dropped together the lower one seemingly refuses to bounce at all, whereas the top one bounces much higher than either ball could on its own-even higher than the sum of the individual bounces.

With less elastic collisions, the optimum ratio for a full transfer of energy is somewhat larger. Spradley determined that there can be a complete transfer of energy as long as the coefficient of restitution is at least .62. If the coefficient is .9, the optimum mass ratio is 3.01. If the coefficient is as low as .62, the optimum mass ratio is 3.24.

A basketball and a baseball have a mass ratio of about 4. When they are dropped as Spradley recommends, the baseball receives nearly all of the basketball's energy and bounces moderately high, and the basketball hardly rebounds at all. A basketball and a Wiffle ball have a mass ratio of about 28. Since the mass ratio is so much larger than in the case of the baseball, the Wiffle ball probably receives much less energy from the basketball than the baseball does. Yet the Wiffle ball takes off like a rocket, climbing higher than the baseball does. (Of course, the elasticity is also likely to be different in the two demonstrations.)

When three balls are dropped and the collisions are perfectly elastic, what should the mass ratio be between the second and third ball for a full transfer? Can you extend the analysis to even more balls? If the mass ratio between the bottom ball and the top one in a large stack is given, can you determine what masses the intermediate balls should have in order to attain the maximum energy transfer? I don't think anyone has yet worked out the answer.

Sometimes I find that certain balls do not bounce as I expect they will. To cite one example, a very small Super Ball should bounce quite high when it is dropped with a basketball, but often it does not. Why not?

In 1986 D. Rae Carpenter, Jr., David J. Rehbein and Robert J. Bonometti, all of whom were then working at the United States Military Academy at West Point, devised a handy way to launch a stack of two balls. In their scheme a lightweight plastic ball that had been removed from a roll-on deodorant dispenser was put on top of a much heavier steel ball of similar dimensions. Then the balls were placed in the top of a long plastic tube, supported by a paper clip that ran through the sides of the tube. When the paper clip was pulled out, the balls dropped down the tube well aligned. Holes had been drilled along the length of the tube so that air could easily escape as the balls fell. The tube was placed on a hard ceramic or tile floor. The mass ratio of the balls was about nine and the coefficients of restitution for the collisions between ball and floor and between ball and ball were high. The plastic ball would usually shoot up to four or five times its release height.

In 1982 an independent analysis of a bouncing stack of balls was published by R. H. MacMillan of the Cranfield Institute of Technology in England. He described a commercially available toy that works like the bouncing balls. The toy consists of a vertical rod on which three cylinders slide. The cylinders are of different lengths, so that they differ in mass; they are stacked in order of decreasing mass. When you pull the cylinders partway up the rod, separate them slightly and then release them together, they bounce off the base at the bottom of the rod and the top cylinder (the lightest one) is shot so high that it flies off the rod.

Bibliography

ENERGY TRANSFER IN ONE-DIMENSIONAL COLLISIONS OF MANY OBJECTS. John B. Hart and Robert B. Herrmann in American Journal of Physics, Vol. 36, No. 1, pages 46-48; January, 1968.

VELOCITY, MOMENTUM, AND ENERGY TRANSMISSIONS IN CHAIN COLLISIONS. James D. Kerwin in American Journal of Physics, Vol. 40, No. 8, pages 1152-1157; August, 1972.

SUPER BALL PROBLEM. G. Stroink in The Physics Teacher, Vol. 21, No. 7, page 466; October, 1983.

VELOCITY AMPLIFICATION IN VERTICAL COLLISIONS. Joseph L. Spradley in American Journal of Physics, Vol. 55, No. 2, pages 183-184; February, 1987.

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