The Mechanics of the Game of Croquet
By C.R. Calladine and Jaques Heyman
Analysis of the game of croquet affords a fascinating exercise in the application of the elementary laws of impact and motion.
The elementary laws of impact will be used to study the motion of croquet balls. An exact analysis is not attempted; assumptions are made which simplify the working, and which lead to results indicating general patterns of behaviour. In calculating numerical values, it will be assumed that the head of a croquet mallet weighs 3 lb, and that the ball weighs 1 lb. A ball has diameter 3 5/8" and the clearance between the uprights of the hoops (competition standard) is 3 11/16".
Hitting the Ball
It will be assumed that the mallet head is moving horizontally at the moment of impact with the ball, and that of the whole mass of the mallet concentrated in its head. If the mallet head is moving with velocity U just before impact with a stationary ball, Figure 1(a), the velocity after impact being V and that of the ball, v,
Figure 1(b), then
where k is the mass ratio of the mallet to ball. It is assumed that the coefficient of restitution, e, is unity; the results may be modified suitably if the impact is imperfectly elastic (see below). It is also assumed in deriving equation (1) that the mallet is moving freely; that is, it is not restrained by the striker.
The ball moves off with linear velocity v and no initial angular velocity; friction with the ground causes the ball to stop slipping, and a rolling motion ensues. The ball is eventually brought to rest by the resistance of the ground to the rolling of the ball. It is reasonable to suppose that the grass does work in bending each blade of grass, etc., over which it travels; the total amount of work done is proportional to the distance travelled. If we neglect the initial slipping motion of the ball we deduce that the distance travelled by the ball is proportional to its initial kinetic energy, and therefore also to .
Equations (1) are consistent with the mallet swinging like a free pendulum. It is possible of course for the striker to force the swing of the mallet to some extent by gripping the shaft with one hand near the head, etc. The limit to the "push" shot corresponds to the velocity of the mallet remaining unchanged through impact: this corresponds mathematically to a mallet of infinite mass. Putting k tends to infinity in (1),
It is not necessary, and indeed it is a foul, to use "push" to control the motion of a single ball: when the striker has to deal with two balls at once, however (the "croquet" shot, see below) the degree of push becomes important.
The Jump Shot
An important shot occurs when the ball is struck by the mallet inclined at a small angle, , to the horizontal, Figure 2(a). Assuming the ground to be perfectly elastic, the ball will rebound with an initial vertical velocity , where v is given by equations (1). This rebound can also be a source of embarrassment to the player who executes a careless roquet shot, since the struck ball may jump over the target ball. The horizontal velocity imparted to the struck ball is
where is the instantaneous coefficient of sliding friction between the ball and the ground.
The great importance of the jump shot is that it imparts (forward) angular momentum to the ball: in Figure 2(b).
where r is the radius of the ball.
From equations (3) and (4)
Forward spin is very useful in hoop running (see below).
If exceeds unity, the linear velocity of the ball will increase above v' when contact is made with the ground after the jump. The jump shot may be used in conjunction with the croquet shot to make the striker's ball overtake the croqueted ball (the pass roll, see below).
A roquet is made when the struck ball hits another ball at rest.
In an attempt at roquet from a long distance, accuracy of aim is the overriding consideration for the striker. With roquets made at short distance, however, it is possible to aim the ball a little off-centre if necessary to move the roqueted ball to an advantageous position on the court (the rush shot). The rush is one of the most important shots in the game: the mechanics of the roquet are thus of interest.
In Figure 3(a) the moving ball strikes a ball at rest so that, on impact, the line joining their centre makes an angle with the direction of motion. Assuming that the coefficient of restitution between the balls is unity, the roqueted ball moves off at an angle with speed , Fig 3(b), the striker's ball being deflected through an angle and its velocity reduced to . The angle is right.
Figures 3(c) and (d) show how the speed and direction of motion vary with the distance of the centre of the roqueted ball from the line of travel of the centre of the striker's ball. These diagrams give an idea of the accuracy of aim and the force of shot required to achieve a rush of the specified distance and direction.
In a straight roquet, , the moving ball stops dead instantaneously, and the roqueted ball moves off with linear velocity v. If the roquet is long, then the striker's ball will have acquired angular momentum, which is not destroyed on impact with the ball at rest; after having stopped instantaneously it will therefore start to roll slightly in the same line. This effect has been neglected in this analysis.
The Straight Croquet
After making a roquet, the striker lifts his ball, places it in contact with the roqueted ball where the latter comes to rest (or on the yard line if the ball has crossed it) and plays a shot on his ball with the balls in contact so that both balls move. This is the "croquet" shot. When the swing of the mallet coincides with the line of centres of the balls, the shot is a straight croquet.
Quoting from the Laws
In the striker makes a foul if he: 1. Push or pull his ball when in contact with another ball without first striking it audible and distinctly and 2. Push or pull his ball when not in contact with another ball, whether he first strike it audibly or not.
[Editor: this Law has been replaced - Law 28a7]
The object of the Law is clear. The striker is not allowed to hit his ball more than once, and gross pushing is disallowed. However, Association croquet is virtually unplayable unless a push croquet shot is permitted, and the Law attempts to give a semblance of respectability to what, in any other game, would be a patent foul.
A wide variety of effects may be produced by variants of the croquet shot ranging from the "stop" to the "push" croquet with an interpretation of shots in between.
For the purposes of analysis three specially simple idealised croquet shots may be distinguished. In the idealised stop shot the mallet is arrested immediately the impact is made: such arrest is, of course, impossible, but the idealisation will indicate a limit to the behaviour of the balls after impact. For the ordinary croquet it will be assumed that the mallet is unrestrained by the striker, and swings as a free pendulum, while the idealised push croquet it will be assumed that the mallet is maintained in motion with a constant velocity U.
It is perhaps worth noting that the idealised push shot and the idealised stop shot are "ordinary" (free-swinging) shots with artificial mallets of infinite and unit mass-ratio ( k) respectively: the infinite range of shots in between corresponds to "ordinary" shots made with an infinite range of mallets.
(a) The Stop Croquet. Figure 4 shows the three successive stages of motion if the mallet head is arrested immediately after striking the first ball. Finally, the striker's ball remains at rest, and the croqueted ball moves off with the velocity v.
(b) The Ordinary Croquet. Figure 5 shows the four successive stages of motion, where V and v are given by the equations (1). In Figure 5(c), the struck ball is instantaneously at rest, and must therefore be subjected to a second impact by the mallet. The final velocities of the balls bear the ratio
It will be appreciated that impact time between mallet and ball, or between ball and ball, has been assumed to be infinitesimal. In fact, of course, the time of impact is determined by the velocity of the stress wave in the balls. In the last paragraph it was said that the struck ball is subject to a second impact; however, there is no question of a foul under the laws. For a very small, but finite, impact time, the pressure between the mallet head and the struck ball will vary, but contact will be maintained.
(c) The Push Croquet. Figure 6 gives the stages of motion; the final velocities of the two balls are equal.
Thus the three main croquet shots lead the striker's ball at rest, moving more slowly than the croqueted ball (in the ratio ), or moving at the same speed as the croqueted ball. By using different amounts of arrest or push, the striker can vary the effective value of k between a few percent to 100% of the distance travelled by the croqueted ball. Further, by using a jump shot combined with push to give the striker's ball initial angular momentum, a shot (the pass roll) may be made in which the striker's ball overtakes the croqueted ball.
The Split Croquet.
The croquet shot most commonly played is the split croquet; in this shot, the striker's ball and croqueted ball are required to travel in different directions and with different velocities. In the following analysis, the line of centres of the two balls at rest makes an angle with the direction of motion of the mallet (Figure 7(a)), so that the croqueted ball moves off at an angle with the line of stroke and the striker's ball at an angle .
(a) The Stop Croquet. Figure 7 shows the successive impacts; Figures 7(b) and 7(c) are identical with Figures 3(a) and (b) for the roquet. Denoting the velocity of the croqueted ball and that of the striker's ball then.
(b) The Ordinary Croquet. Two cases arise. If is large, the croqueted ball will offer little resistance to the striker's ball, and will be small. This is the "take-off" and the motion is illustrated in Figure 8. Equations (6) hold, where v is given by equations (1).
If is small, the mallet will make a second impact with the striker's ball; from Figure 8(c), the condition for this to occur is
on substitution from equations (1). If inequality (7) is satisfied, Figure 9 illustrates the second impact on the striker's ball.
In Figure 9(b),
(c) The Push Croquet. Figures 10(a), (b) and (c) illustrate the push take-off croquet;
From Figure 10(c), a second impact occurs if
If inequality (11) is satisfied, the final velocities of the balls are shown in Figure 10(d):
Equations (12) confirm the empirical rule for small angle push splits, that the line of stroke should bisect the angle between the required lines of travel of the two balls.
Figure 11 illustrates the results so far. The full lines show the relation between the angle and ; the figure shows that croquets can be made without a second impact occurring on the striker's that ball if is greater than 45° for a push shot, or is greater than 35.3° (approximately) for an ordinary shot.
Imperfectly Elastic Impacts.
It has been assumed in all the above working that no energy is lost on impact. By introducing a coefficient of restitution, e, the analysis may be repeated without difficulty although with some tedium. Equations (1), for example, become
and equations (2) become
In Figure 11, for the split croquet, are shown dashed curves calculated for e = 3/4. The results for ordinary and the push croquet are little affected, and the angles dividing the take-off from the croquet with second impact are altered by very little, from 45° to 45.6° and from 35.3 to 36.1°. The curves connecting and for the stop croquet shows a marked change of behaviour for small ; the perfectly elastic curve, , has a singular point at the origin. However, these results for small are hardly significant for the stop croquet, since it has been assumed that the mallet is stopped immediately after impact.
Assuming that the distance travelled by a ball is proportional to the square of its initial velocity, Figure 12 shows the results presented in terms of distance travelled against angle of split for e = 3/4. For a straight croquet, the striker's ball travels about a third of the distance of the croqueted ball, if an ordinary stroke is made. Empirically, in an "ordinary" croquet the corresponding distance travelled is one-third to one-half, so that, if the above analysis is correct, the mallet is not moving freely, and some "push" is applied to the balls, corresponding to an "ordinary" croquet with an artificial mallet with k approximately 4.5. Push is inevitable for any but the shortest croquet; the back swing of the mallet is, relatively, so small that considerable acceleration has to be given to the mallet in order that the required velocity U can be reached at the moment of impact. The force required for this acceleration will act throughout the impact and, since the impact lasts a small that finite time, extra energy will be imparted to the balls, typically to that of the striker.
In a straight croquet, it is usually the striker's intention to send both balls to precise locations on the court. For a small angle of split the croqueted ball can be placed most accurately. The balls are aligned with their lines of centres along the intended path of travel, and the croqueted ball will travel almost exactly along this line independently of the angle of the stroke. The strength of stroke can also be gauged accurately so that the croqueted ball travels the correct distance.
The behaviour of the striker's ball is however more difficult to control. For example, suppose that the final destinations of the balls are such that a 45° angle of split is required, and that the striker's ball is to travel three-quarters of the distance of the croqueted ball. Figure 12 indicates that an ordinary shot will send the striker's ball about 0.45 of the distance travelled by the croqueted ball; for a push shot, the ratio is over unity. Some degree of push is evidently required to achieve a ratio of 0.75, and the direction of stroke depends on the amount of push applied. Figure 13 gives the same curves as Figure 12, but replotted to show angle of split against angle of stroke . For the 45° split, lies between about 17° and 23° of the ordinary and push shots respectively. If the strokes were made at = 17° and a full push applied, the angle of split would be about 33° instead of the 45° intended; conversely, were an ordinary stroke made at = 23°, the angle of split would be about 57°.
If push, and particularly roll, is applied to the striker's ball, the path of the croqueted ball will not be perfectly true. Peeling (intentionally putting the croqueted ball through its correct hoop) is for this reason difficult from a split croquet. The drag of the striker's ball on the croqueted ball will pull the latter so as to close the angle of split. The margin of error in hoop running is so small (see above) that a roll split croquet combined with a peel should be avoided if at all possible. A stop split croquet, on the other hand, seems to impart negligible pull to the croqueted ball.
In the take-off croquet (large angle of split), the striker's ball can be placed more accurately than the croqueted ball; frequent use of the take-off is the mark of the high bisquer who is frightened of controlling the small angle split. The direction of travel of the croqueted ball can again be set accurately, but the distance may be limited by the rule which forbids either ball to cross the boundary of the court during the croquet stroke. Thus, if the angle of split required is near 90° (it cannot, of course, the greater than 90°), the croqueted ball will move very little (it is required by the rules only to tremble) while the striker's ball travels the full length of the court.
Successful hoop running depends almost entirely on imparting sufficient forward roll to the ball.
The clearance between ball and hoop is so small that it is impossible for a ball to pass a hoop without at least touching an upright from an angle of more than 8° with the centre line of the hoop. The accuracy required to run the hoop freely from a point as near as 12 inches to the hoop on the centre line is formidable: 1/32" deviation in 12 inches corresponds to an angle of less than one quarter of a degree, or to making a roquet over the full length (35 yards) of the court. If the ball approaches the hoop along the centre line with an eccentricity slightly larger than 1/32" it may pass the hoop after rebounding from one of the uprights. For a slightly greater eccentricity, however, the ball will undergo a series of bounces back and forth between the uprights and, in so doing, may lose its forward momentum and come to rest in the jaws of the hoop. In this process, however, some of the angular momentum may have been conserved and this may carry the ball forward after its momentary arrest.
For a straight or nearly straight hoop run, the usual shot is slow and deliberate. Great care is taken with the aim and the ball is accelerated slowly so that it may acquire the angular momentum necessary to carry through in the event of collision with the uprights. This type of shot, although common, is a foul in that the ball is almost certainly pushed.
A legal method of obtaining high angular momentum is by using a deliberate and slow version of the jump-shot made with the mallet tilted forward.
For wide-angle attempts at hoop running where the ball must necessarily hit one of the uprights almost head on, the jump-shot is useful. If the ball hits the uprights above ground level, there is no occasion for friction from the ground to reduce the angular momentum of the ball: thus, when the ball drops to the ground it may successfully pass the hoop. The shot, well played, enables hoop to be run from apparently impossible angles.
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