Newton's Cradle in Billiards and Croquet
Author: Dr Ian Plummer
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Phys. Educ. 51 (2016) 065020 (4pp)

Rod Cross

Physics Department, University of Sydney, Sydney NSW, Australia

E-mail: rodney.cross@sydney.usyd.edu.au

[Manuscript supplied by the author]

Abstract

When an object collides head-on and in line with two balls in contact, the outcome is not generally easy to predict. We consider three simple examples. One is Newton’s cradle with only three balls. Another is a billiard cue colliding with the two balls. The third is a croquet shot where a mallet collides with the two balls. The outcome in each case is different since it depends on the mass of the colliding object.

Online supplementary data available from https://iopscience.iop.org/article/10.1088/0031-9120/51/6/065020/data

1. Introduction

Newton's cradle is so well known that it needs no introduction, except to note that it was first described and explained in 1671 by Mariotte [1]. The inventor of the modern version used Newton’s fame to help sell his product. When using only three balls, the outcome is easy to explain in simple terms [2] although the finer details are quite complex [3]. If one ball collides with two at rest then the incident ball comes to rest, the middle ball remains at rest and the ball at the far end exits with essentially the same speed as the incident ball. The result can be described at a simplified level by assuming that the balls initially at rest are slightly separated. When the incident ball strikes the middle ball, the incident ball comes to rest and transfers its energy and momentum to the middle ball. The middle ball then strikes the end ball and transfers its momentum and energy to the end ball. If the two balls at rest are actually touching then none of the balls comes to a complete stop but the exit speed of the end ball is much larger than speed of the other two balls [2, 3].

The outcomes are illustrated in figure 1(a) and are demonstrated in supplementary videos 4589 and 4590 using three billiard balls (media).

The author was recently asked, by a keen croquet player, to explain the physics of the croquet shot. In that case two balls are placed in contact on the ground and the player strikes only one of the balls with a mallet, sending both balls off at different speeds and possibly in different directions. If the path of the mallet is in line with the centres of the two balls, then the shot is called a straight croquet and both balls head off in the same direction. The specific problem faced by the player was how he can best vary the ratio of the speeds of the two balls. Not knowing anything about croquet, I consulted YouTube and discovered that the secrets were in placing the feet behind or in front of the balls to vary the contact angle of the mallet and in placing the hands near the top or bottom of the handle. In addition, posture is important. The web site at www.oxfordcroquet.com contains many technical articles on the physics of croquet but none address the specifics of varying the ball speed ratio.

To investigate further, I set up a simple experiment using two billiard balls and a billiard cue as an analogue of the croquet problem. The outcome is shown in figure 1(b). When the back ball is struck by a cue, the two balls take off with the same speed, regardless of whether the back ball is struck below its centre to impart backspin or above its centre to impart topspin or struck in the middle to impart zero spin. The behaviour is quite different from Newton's cradle, but can be explained in the same way. That is, we can assume that the two balls are slightly separated. When the back ball is struck by the cue it takes off at speed v say. When the back ball strikes the front ball, the back ball comes to rest and the front ball takes off at speed v. The cue itself is heavier than the balls and is held firmly in the hand which is attached to a heavy forearm. Consequently, the cue slows down by a negligible amount after the first collision and strikes the back ball again at about the same speed. As a result, both balls head off at essentially the same speed, as demonstrated in supplementary video 4582.

A croquet shot represents an intermediate example of the the first two cases in figure 1. A typical result is shown in figure 1(c) where the mallet speed decreases from V1 to a lower speed after it collides with the back ball. If the two balls are separated slightly before the initial collision then the back ball comes to a stop when it collides with the front ball. The mallet then collides a second time with the back ball, decreasing in speed to V2. As a result, the speed of the back ball after the second collision is generally lower than the speed of the front ball. If the back ball is to remain at rest, then the mallet would need to have the same mass as a ball. If the front and back balls were to travel at almost the same speed, then the full weight of the hands and arms would need to be behind the mallet. In practice, a mallet is typically about three times heavier than a ball, so it is not possible for the back ball to remain at rest after the collision. Nevertheless, it is possible to vary the speed ratio by moving the hands to the top of the handle or by moving one hand towards the bottom of the handle.

2. Experimental Results

Each of the three types of collision shown in figure 1 were recorded on film using a Casio EX-F1 video camera operating at 600 frames s−1. The collisions were examined using two billiard balls resting on felt cloth stretched over a small table. The main qualitative features are shown in the supplementary videos. The results indicated that differences can arise when the two balls at rest are touching rather than being separated, especially when the impacting object is heavier than the two balls. As the separation distance between the two balls decreases, the two collisions occur closer in time. In the limit where the two balls are touching, the two short collisions tend to merge into one longer collision and are not easily identified on video film as being two separate collisions. A heavy object impacting the two balls does not come to rest immediately after those two merged collisions and can make yet another collision with the two balls. To help avoid confusion, we can describe the two merged collisions as the first and second collisions, and any subsequent collision can be described as the third collision, despite the fact that the video evidence suggests that there may have been only two collisions.

For example, when two billiard balls in contact were struck by a cue, there was no visible separation of the two balls immediately after the impact with the cue. Instead, the two balls remained in contact for at least 10 ms or more, separating from the cue about 3 ms after the initial impact. Sometimes, the cue made a third collision with the two balls about 10 ms after the first two (merged) collisions. As a result of the third collision, the two balls separated. An example is shown in supplementary video 4592, best observed one frame at a time, and can be explained as follows.

Suppose that two identical balls are travelling at the same speed v1 and are slightly separated. If the back ball is struck by a cue, suppose that its speed increases to v2 so it catches up to the front ball. As a result of the collision, the front ball speed increases to v2 and the back ball speed decreases to v1, assuming that the collision is perfectly elastic. Consequently, a subsequent collision of the cue with the two balls will result in further separation of the two balls. The third collision of the cue with the two balls in video 4592 arose because the cue was accelerated slightly by hand over a relatively long period. If the third collision is avoided, or is relatively weak, then the result is as shown in figure 1(b). The third collision can be avoided by angling the cue down on the ball instead of striking the ball in a horizontal direction. Alternatively, a weak third collision will arise when using a stop shot where the cue decelerates rapidly just before and after the first collision (as shown in supplementary video 4578).

A simple 430 g wood mallet was constructed to investigate the situation shown in figure 1(c). The mallet was allowed to strike two billiard balls in contact, each of mass 115 g. Unlike the case with the billiard cue, slow motion video showed that (a) the front ball separated from the back ball about 1 ms after the initial impact of the mallet with the back ball and (b) the back ball remained in contact with the mallet for about 20 ms after the initial collision. The back ball did not come to rest since it remained in contact with the mallet and was pushed forward by the mallet. Instead of a second collision, the back ball was simply accelerated by the mallet and separated from the mallet only when the forward speed of the mallet decreased below the forward speed of the back ball. The end result was as shown in figure 1(c), and as shown in the supplementary video 4608, but it was not the result of two separate collisions. Rather, it was the result of an initial collision lasting about 1 ms, followed by an acceleration or push phase lasting about 20 ms.

Similar results were obtained using a smaller wood mallet to strike two superballs in contact. In that case, the duration of the initial collision was about 4 ms since a superball is much softer than a billiard ball. Compression and expansion of each ball could be seen more clearly on the video film. In addition, it was observed that friction between the back ball and the felt cloth on the table acted to oppose the accelerating force of the mallet. Since the duration of the collision was longer than with the billiard balls, and since the mallet used was lighter, it was possible to give the two superballs a short, gentle tap, in which case the back ball and the mallet both came to rest, as indicated in figure 1(a) and as shown in the supplementary video 4618. A more aggressive impact is shown in supplementary video 4632 where compression of both balls can be seen as well as a weak second collision followed by a short push phase. The first two collisions tend to merge into one long collision, although the front ball separates from the back ball before the second collision has ended, the end result being as shown in figure 1(c).

3. Conclusions

As pointed out by Gavenda and Edgington [2], the collision results obtained with Newton’s cradle are consistent with conservation of energy and momentum, but there are many other potential outcomes that would be equally consistent with the conservation laws. The same situation applies to the collisions illustrated in figures 1(b) and (c). For example, if the collisions were perfectly elastic then a potential outcome in both figures 1(b) and (c), consistent with the conservation laws, is that the back ball might remain stuck to the cue or the mallet. The actual outcome is complicated by the facts that (a) real-life collisions are always inelastic so kinetic energy is not conserved and (b) energy is added by the person wielding the cue or mallet during the collision processes involved. Nevertheless, it is satisfying that the end results can be explained in simple terms by assuming that the two balls are separated before the collision, even if they are not. The actual sequence of events when the balls are initially in contact is more complex.

The physics of the collision processes illustrated in figure 1 is also relevant in traffic accidents, for example when a bus or truck collides with two or more cars ahead of it. The basic physics could be investigated further in a student laboratory using gliders on an air track, even without the added information that can be obtained with the aid of a high speed video camera.

Received 6 July 2016

Accepted for publication 3 August 2016

doi:10.1088/0031-9120/51/6/065020

References

[1] Cross R 2012 Edme Mariotte and Newton’s cradle Phys. Teach. 50 206-7

[2] Gavenda J and Edgington J 1997 Newton’s cradle and scientific explanation Phys. Teach. 35 411–7

[3] Hutzler S, Delaney G, Weaire D and Macleod F 2004 Rocking Newton's cradle Am. J. Phys. 72 1508–16

Rod Cross is a retired member of staff at Sydney University and continues to work mainly on the physics of sport.

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