Croquet Drives, Pass-Rolls Stop-Shots and Scatter-Shots
Don Gugan, Bristol
The availability on YouTube of the CA's high-speed croquet video (1) enables anyone interested to see exactly what happens during a stroke: mallet and balls are taking part in a 3-body interaction – and in general these are impossible to analyse. However, it can be seen that every stroke shows three distinct phases – though sometimes overlapping slightly – each lasting about a millisecond, but each involving only two of the three bodies (cf. (2) figure 4): first, the impulse of mallet on the striker's ball, ball 1; second, the almost immediate impulse where ball 1 transmits most of its momentum to the croqueted ball, ball 2 – though retaining a small amount; and third, a double-tap by the mallet as it catches up with the striker's ball and delivers to it a further impulse. How these three impulses affect the speeds of the mallet and balls for the first few milliseconds of a drive is illustrated schematically in figure 1. Detailed measurements have been made on the video images on the original CD to study the physics involved and to find whether treating a croquet stroke as three 2-body interactions in rapid succession can provide a realistic simple model for calculating its outcome, (2), but essential information was missing, and further experiments are reported here which give a better comparison with theory.
Much later than the time scale of the previous impulses, a further and often extended contact between mallet and striker's ball may occur, often also called a double tap – though probably better described as a push. Such pushes are of course always faults, and while sometimes easily visible or even audible, are often difficult to recognise during play and hence frequently overlooked.
A good but no doubt accidental example of a push is shown (albeit in a single ball stroke) for stroke A1R on the video – (at about frame 280, about ten times later than the duration of the actual stroke; cf. (2) section 7 for a more detailed discussion) – but writing a clear and succinct law to outlaw them has caused considerable difficulty, as evident in Law 28(a)(7,8). As is clearly observable on the video images, all straight croquet strokes must be double taps, and have a stage where the striker's ball is squeezed between mallet and croqueted ball (i.e. are extended contacts), both things which, at a time when the full details of a collision could not be seen, the Laws of Croquet (cf. above) defined as faults. By a strict interpretation of the Laws the only legal stroke on a striker's ball in contact with another ball would be to play it away, just as is the case in the game of snooker: but fortunately the founders of our game exempted the taking of croquet from those Laws.
Earlier definitions of taking croquet stipulated that the mallet should not be accelerated through the stroke, which would imply that the speed of the mallet must remain constant – an impossibility, of course – but the intention is reasonably clear, i.e. that after the mallet has been slowed down by collision with the striker's ball, it should continue to move with a constant speed. The stroke that comes closest to fulfilling this requirement is the drive, especially when the mallet is swung freely from the top of the shaft in a 'pendulum' style and falls largely under its own weight. Under these conditions, and in particular when the stroke is slightly 'uppish', i.e. when the mallet is sloping slightly backwards at impact so that the ball is lifted from the ground – which then exerts no friction on the first stages of the resulting motion – one may expect that the model of independent two-body collisions will be a good approximation for the behaviour of the stroke, and in particular that it will be possible to calculate accurately the ratio of the initial speeds of the balls, v2/w1 (cf. figure 1), and what is of more interest, the ratio their final travel distances, L2/L1.
The heavy solid lines in figure 1 show the ideal behaviour of the mallet and balls calculated for a drive from Newton's Laws of motion for point masses (sufficient for the analysis since no rotations are initially involved, cf. (2)), using mallet/ball mass ratio, M/m = 2.30, and coefficient of restitution (CoR), e = 0.772, as discussed further below; these parameters were unfortunately not given in the CA video, so that while plausible values for them could be calculated from the measurements, the comparison with theory was less than ideal.
To improve this situation, further experiments have been made, including both 'pendulum' type drives, trying not to exert any extra force on the mallet shaft during the stroke, and stop-shots made trying to obtain the greatest possible distance ratio between the balls by using a downward 'jab' just before impact, so that friction between the ground and the heel of the mallet stops it soon after impact. The drives were of modest strength, with a mallet speed of about 3m/s sending the croqueted ball about seven yards on a well-grassed, closely-mown and fairly fast lawn; and the stop-shots were of similar strength. The mallet used was an older type hard-faced, square design of weight 3.25lbs: but the relevant mass for the analysis is the mass of the head of the mallet, which was found by taking moments about its point of balance to be 2.30lbs, thus giving a value for the mass ratio, M/m, of 2.30 ± 0.01. The balls were Dawson 'Internationals', all in excellent condition: independent measurements of their CoR were not made, but the makers report that under the standard bounce test '… International balls rebound … 35½" – 36" …' (Bryan Dawson, private communication) corresponding to a CoR, e = 0 .772 ± 0.003. From these values one can calculate (cf. (2) equations (a8), (a9)) that the initial ratio of ball speeds after a drive is equal to 2.03 ± 0.01; and, making the usual assumption that the balls finally come to rest after their kinetic energy has been dissipated by a uniform friction force as they roll over the grass, the ratio of their distances of travel, L2 and L1, is then given by the square of the speed ratio, i.e.
The results for twelve each of drives and stop-shots are shown in figure 2 below.
The dozen values of distance ratio shown in figure 2 have a mean value of 4.23 ± 0.2 (where the uncertainty is the standard error of the mean), i.e. negligibly different from the calculated value of 4.12 ± 0.04. The experimental scatter, i.e. the standard deviation of individual strokes, however is 0.6, about ten times the random error of the measurements, and reflects the reproducibility of the separate strokes, clearly a matter of individual skill but which any systematic experimental study of the physics of croquet strokes needs to take into account. The drives shown on the CA video are of little help here: stroke C1D is the only one in the same range as the present data, and gave a ratio of 4.0 at distance 6.21m; the agreement is good so far as it goes, but the video includes no repetition of similar strokes, so the precision of the single datum is unknown – and the stronger drives on the video, C3D and C10D, had distance ratios of 3.0 and 3.8 for L2 equal to 16.17m and 30.80m respectively – which does not suggest a scatter notably less than that reported here. There is a suggestion in the data that the distance ratio is tending to fall with increasing values of L2, i.e. stronger strokes, which seems not unlikely, but the scatter does not allow any firm conclusion to be drawn without a much larger data set, or preferably with a much more consistent stroke – possibly requiring some mechanical device.
The present results lead to the conclusion that the 'pendulum' drives are well described by the model where the mallet continues to move with a constant speed after the impulse it has given to ball 1, i.e. between the times B and E shown for the solid red line in figure 1: however, other possible sorts of croquet stroke are illustrated by the dashed red lines in the figure.
The thin dashed line indicates a hypothetical possibility where the mallet has been brought to a stop before the time E so that it never overtakes ball 1 and the third impulse I3 does not occur. In this case, ball 1 would continue to move with speed v1* its residual speed after I2, the ball on ball impulse, and as a result the distance ratio L2/L1 would have a limiting value of (v2/v1*)2, dependent only on the value of e for the balls (cf. (2) equation (2)), and in the present case equal to 60.4. The distance ratios discussed below for the 'jab'-type stop-shots, the nearest one can get to the hypothetical case, are nothing like as large as this – and considering the movement of the balls that must occur even in contact times as short as a millisecond, it is hard to imagine how for touching balls they ever could be: for straight croquet drives the inevitable conclusion from the physics of the stroke is that 90% of the energy and distance of the striker's ball comes from the double-tap I3.
The thick dashed red line for the motion of the mallet in figure 1 shows a stroke where, after its initial impact, the mallet is forced to regain its original speed, U. The second impulse of the mallet on ball 1 is now almost as strong as the first, and the resulting speed of ball 1 rises to 1.202U compared with 1.235U, the speed after the first impact, v1, and significantly greater than the speed of the croqueted ball, v2 = 1.094U , cf. the heavy, black dashed line in figure 1; as a result ball 1 here travels further than ball 2 (provided that there is at least a small angle of split to avoid a later collision), giving a L2/L1 ratio of 0.83. Distance ratios near unity are normally the province of the full-roll and achieved not by control of mallet speed, but by angling the mallet to put downward pressure on the striker's ball at impact in order to produce controlled amounts of initial rotation, cf. the full-roll C10F on the CA video where L2/L1 = 1.24, while ratios of less than unity are pass-rolls, cf. the ratio of 0.94 for C10P – though the two other nominal pass-rolls on the CA video are not good examples since both had ratios greater than unity, i.e. they were not 'passes' at all. It appears that distance ratios of about unity are theoretically possible in straight drives, without the need to put roll on the striker's ball. In fact, some players are adept at producing surprisingly large amounts of 'pass', sending the striker's ball about twice as far as the croqueted ball in powerful straight shots – 'super-pass-rolls' perhaps. Precisely what happens in these strokes seems never to have been examined, and though their legality has sometimes been impugned as a 'push', experienced referees have seen no clear reason to fault them. They always seem to involve a crouching stance and a sweeping action with a lot of arm follow through, and it seems likely that this is necessary to produce a strong second impulse on ball 1 such as that discussed above, but if, in addition, there is also a forward inclination of the mallet shaft to produce the downward pressure which induces spin, then it is likely that both effects will be active in producing an extreme amount of pass, without any need for a later and illegal 'push'. One can estimate an upper limit on a pass-roll from the equation which allows for the effect of any initial spin of a ball on its subsequent linear rolling speed (cf. (2), equation (D1)),
assuming that immediately after the croquet stroke, time F in figure 1, both balls have the same linear speed, v', and that the striker's ball 1 has its maximum amount of initial forward spin consistent with not slipping on the grass, so that Ω1 = 1, whereas the croqueted ball 2 has negligible initial rate of spin, Ω2 = 0 (which is in fact always the case); then one finds from the equation above that the initial rolling speed ratio after allowing for the contribution of spin is v2/v1 = 5/7, and thus that the extreme distance ratio,
i.e. the striker's ball goes almost exactly twice as far as the croqueted ball – which seems from casual observation to be about the limit of these 'super pass-rolls'. A high speed video of the initial impact might be revealing, and no attempts have been made to test this limit, but no separate 'push' need necessarily be involved in the stroke, and it appears to lie within the range of legality.
As can be seen from figure 2, while the distance ratios are far less than the limiting value of 60.4 given above, they are significantly higher than for drives – as indeed intended, but with a much worse reproducibility, both in the ratio, and also in the strength of shot as shown by the value of L2. The mean value of the ratio for the twelve strokes is 7.15 ± 0.6, but the scatter is about ± 2.0, which makes it clear that attempting high stop-shot ratios using a 'jab' technique is unreliable – and also that the value of the mean has no particular significance.
While ratios of ten or more are certainly possible, the results suggest that more modest ambitions will be preferable: the green line in figure 2 passes through the mean of the lowest seven points, 5.64 ± 0.6, probably a more realistic target for most players; this corresponds to a mallet speed of 0.400U for the second tap, compared with 0.463U for the unrestricted swing – the 'jab' does not seem a very effective technique for decelerating the mallet, as well as being a hazard to the lawn: even for the best 'stop', with L2/L1 = 11.6, the mallet speed has only been reduced to 0.295U. The CA video shows three stop-shots, C3S, C10S, and C20S, with distance ratios 6.6, 4.2 and 4.1 corresponding to L2 values of 4.85m, 10.61 and 20.26m respectively; the first of these is very close to a value measured here, but, as it happens, the stronger shots are not distinguishable from ordinary drives: whether this is an inevitable consequence of playing a stronger stroke would be interesting to know, but it was not explored in these experiments.
Whether a 'jab' can put a back-spin on the striker's ball and thus decrease its travel (cf. the equation above with a negative value of Ω) has not been systematically studied, and though the CA video reveals back spin, it is only at the level of about 10%, much less than the value Ω ≈ -1 needed to produce a considerable effect.
The previous results and discussion make it plain that, quite apart from faults arising from blatant pushes, any stroke into touching balls will give rise to a double-tap and an extended contact, and the same faults are highly probable even if the balls are a few millimetres apart. The only sure sign that no double-tap has occurred is when the scattered ball can be seen to travel much further than the striker's ball – and for any ball which conforms to the CA bounce test 'much further' means, for straight shots, at least fifty times as far. Angled shots are harder to judge, but even for a stroke at 45° to the line of centres, the ratio of the distances in the direction of ball 2 only decreases to about twenty-five: if these ratios are not reached, the impulse I3 of figure 1 has occurred, and the extra distance of the striker's ball is the result of a double-tap.
When the balls are not in contact a double-tap is not inevitable, but, dependent on the gap between the balls and the strength of the stroke, one will often occur (cf. (2), figure 4). The Commentaries to the Laws recognise the problem, and suggest minimum values of gap needed to avoid a fault, e.g., 2mm for croquet, and 4mm for 'golf'; in fact, these figures only draw attention to the likelihood of a fault, but fail to emphasise the necessary and unmistakeable sign of a fault-free stroke, i.e. that the resulting distance ratio must be very large; fifty, as discussed above. The Commentaries to the Laws of Croquet suggest that in a 'clean' stroke ball 1 should travel "significantly less" than ball 2 – which hardly seems to meet the case – e.g. is half the distance significant, or a tenth?; while for the game of 'golf', C13.1, Law 13(a) (6, 8) rules that a ratio of more that eight signals a 'clean' stroke; a value of eight is certainly tending towards a more realistic level, but figure 2 shows that five out of twelve stop-shots which were indubitable double-taps nevertheless had distance ratios of greater than eight – a further factor of six is still needed to achieve reality.
For scatter-shots on balls which are touching, the confusion is worse; they seldom arise, but when they do, players (even referees) are often unaware that the only legitimate shot is to play the striker's ball away, without moving the other ball, as in snooker, as mentioned before. This ought to be unmistakeably clear for croquet, but the opacity of Law 28(a)(7,8) means that it is not; while for 'golf', the Law, though clear, is perverse: after stressing the likelihood of faults for small separations and the difficulty they cause to referees, C13.1 Law 13(a)(8), rules that "unless the two balls started in contact"… "all strokes… into balls 4mm apart or less will constitute a fault": i.e. for separated balls a likely but by no means certain double-tap is declared to be a fault, but for balls in contact an inevitable double-tap is declared to be 'clean' – a bizarre ruling, fully deserving Mr Bumble's opinion of the law. Some amendments seem to be in order.
(1). https://www.youtube.com/watch?v=M4EH7Omxkc8 Single ball shots
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