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Technical
Experiments on the Game of Croquet; Bouncing and Rolling Balls

By Don Gugan

1. Introduction

The experiments described here were carried out about twenty years ago, the motivation being Law 3e(3) which states that "A mallet head ..... may be made of any suitable materials, provided that they give no significant playing advantage over a head made entirely of wood". As most players used heads not made entirely of wood it was evident that they preferred them to wood, but whether they had any playing advantage, or even whether they behaved differently, was not clear. One way of finding out was to measure the coefficient of restitution for a variety of balls and mallet faces in common use, and for a range of impact speeds so as to simulate different single-ball strokes.

Some preliminary trials were made on lawns to find the range of ball speeds relevant for the impact experiments. These are discussed in an appendix, together with an analysis of rolling and sliding motion, and with suggestions for improved lawn speed tests.

2. Bounce Tests and the Coefficient of Restitution

2.1. The theory of linear collision: The speed given to a ball after striking it with a mallet depends on the mass of the mallet, M, on the mass of the ball, m, and also on the energy loss during impact, which is characterised by the coefficient of restitution, e. The value of e is defined as the ratio of the speed of separation of ball and mallet after impact to the speed of approach before. For ideal elastic behaviour e is equal to one, but for a real, inelastic impact e is less than one, and the energy given to the ball is reduced in proportion to (1 - e2).

The equations for linear elastic collision between mass M moving with initial speed Vi striking a mass m initially at rest (i.e. vi = 0), so that after collision m acquires a final speed of vf, while the speed of M is reduced to Vf, are (see e.g. Daish [1], pp 103 - 105),

vf = Vi[2k / (k+l)];      Vf = Vi[(k-1) / (k+l)],
(l a,b)

where k is the mass ratio M/m. For a croquet mallet and ball, k is usually close to 3, when the value of the ball speed becomes vf ~1.5Vi, while the mallet speed falls to about half its initial value. When the coefficient of restitution is also included in the solutions to the linear equations, the expressions for vf and Vf become,

vf = Vi[k(l+e) / (k+l)];      Vf = Vi[(k-e) / (k+l)],
(2 a, b)

and it is easy to confirm from equations (2) that the final speed of separation of M and m after collision is eVi. In order to measure e it is convenient to drop a ball onto a very large test surface at rest (i.e. M is essentially infinite), when the initial speed of approach just before impact, U, gives rise to a speed of separation eU immediately after impact as a result of the inelastic collision.

The Croquet Association (CA) specifies the range of values of e for a ball to be acceptable: a ball dropped from a height of 60" onto a firmly anchored steel plate must bounce to a height of between 30" and 45", i.e. for impact speed of 5.47 m/s, a value typical of fairly strong strokes, e must lie between 0.71 and 0.87 (see also later).

2.2 Drop tests: The coefficient of restitution, e, was measured by vertical drop tests onto several surfaces, much as for the standard CA drop test. However, the height of the drop was also varied so as to give initial impact speeds between 2 and 6.5 m/s, and e was determined from measurements of the ball speed just before and just after impact.

A test anvil was made from a cylinder of steel weighing 70 kg (about 150 times the mass of the ball). The ball was bounced directly on the steel surface for experiments on a ball-steel collision, and the anvil was also provided with a ring of six bolts so that mallet heads could be firmly clamped to it by pressing down on a rod which passed through the shaft hole. The ball was released through a guide tube which located it at the drop height and so that it fell without rotation within ±1mm of the centre of the test surface. The ball interrupted a horizontal laser ray as it fell, and this triggered an electronic timer (see below) which recorded the transit time of the ball and allowed its initial mean speed at the ray height to be calculated. On bouncing, the ball again interrupted the laser ray, giving the speed after impact. For lower impact speeds, between 0.5 and 2 m/s, the anvil was turned on its side and the ball swung by a long, light, bifilar pendulum so as to make impact at the bottom of the swing. The apparatus is shown in figures 1 and 2. The maximum speed studied was only half that needed for the strongest strokes in croquet (cf. appendix), but was as high as can conveniently be obtained without the use of a gun of some sort to project the ball at the test surface.

construction of anvil for ball drop experiments
Fig. 1: Schematic diagram of drop mode impact tests.
pendulumn landing on anvil
Fig 2: Schematic diagram of pendulum mode impact test (anvil of figure 1 laid on its side).

2.3 Timing system: The beam from a laser was collimated so that a ray ~1 mm wide fell on a phototransistor. The off/on pulses from the phototransistor as the ball fell through the ray were sharpened by a timer (i.c. type 555), then fed into a five stage Johnson counter (i.c. type HEF4017BD) connected so that the two pulses as the ball fell triggered a crystal controlled Racal counter (type 535a) giving the fall time tf, while the pulses as the ball rose triggered a second Racal counter, and gave the rise time tr, The ratio of these two times, r = tf/tr, gives an approximate value of e, however, there is extra gravitational acceleration between the ray height and the plane of impact which must be corrected for, giving for the vertical drop tests,

e = r{[1 + x(tf/r)2 + y(tf/r)4] / [l + xtf2 + ytf4]},
(3)

and for the pendulum swing tests,

e = r{[1 + z(tf/r)2] / [1 + ztf2]},
(4)

where the values of x, y and z are given by the gravitational acceleration, g ( = 9.81 m/s2), and by the dimensions of the apparatus shown in figures 1 and 2,

x = g(h - R) / 2R2;     y = (g / 8R)2;      z = (g / 4L).
(5 a, b, c)

The initial impact speeds can be calculated from the values of tf, suitably corrected for the extra acceleration and it was confirmed that they agreed closely with values expected for free fall under gravity from the initial fall heights, H, i.e. air-damping of the motion was negligible. The initial speeds at impact, U, for the drop tests and the pendulum tests respectively, are thus,

U = (2gH)1/2,     and U = X(g / L)1/2.
(6 a, b)

The whole timing system was tested by a simple pendulum experiment, and was accurate and internally consistent to about 0.1%, with a resolution of 10 µs. In these experiments the ball swung symmetrically across the laser ray, and the timers were triggered to measure the half period of oscillation, and also the transit time of the ball, which allows the amplitude of oscillation to be calculated. The Johnson counter reset automatically after each tenth pulse and so gave continually updated times over hundreds of oscillations. Good agreement was found with established results for the dependence of period on amplitude, the exponential damping due to air drag, and the change of damping with amplitude (cf. Gupta et al, [2])

Besides the coefficient of restitution measurements, the area of contact was obtained from impressions made on a piece of carbon paper or thin foil, and the duration of contact was measured by adapting the timing system so that the start and stop pulses were produced by the make and break of electrical contact of the impact surfaces, which were each covered by a thin layer of aluminium foil. These measurements are not discussed further here, but they give information on the nature of the inelastic impact which is discussed in [3].

3. Experimental details

Experiments were made on four well-used Jaques "Eclipse" balls, two Walker balls, and two Townsend balls, which will be identified as J, W, and T balls respectively. W and T were balls in use for club play at the time of these experiments but which have not gained general acceptance amongst players, however, their behaviour is instructive. The balls were dropped on three surfaces; on steel, much as in the CA drop test; on wood, using a round head typical of older style mallets, but with faces in very good condition; and on plastic, using a new head made from a solid piece of plastic believed to be PTFE.


 
Impact speed, U(m/s):
COEFFICIENT of RESTITUTION, e(U) x 100
Pendulum Tests
Drop Tests
0.50
1.00
1.64
2.04
2.00
2.83
3.51
4.43
5.47
6.01
J BALL, on Steel:

84.8

82.9

81.6

80.7

80.9

79.5

78.4

77.3

76.3

75.7*

on Wood:

85.7

84.2

82.9

82.4

82.7

81.7

81.1

80.6

79.9

79.5

on Plastic:

-

85.6

83.3

81.9

81.3

79.1

77.1

75.7

74.3

73.5

W BALL, on Steel:

88.7

88.5

88.7

-

85.5

85.5

86.1

85.5

85.1

85.5

on Wood:

89.3

90.2

90.3

89.7

88.8

88.8

89.9

89.5

89.5

89.2

on Plastic:

88.6

85.0

81.0

78.6

78.2

75.3

72.5

70.7

68.2

67.0

T BALL, on Steel:

96.9

97.0

96.6

-

96.5

95.9

95.3

93.0

91.2

89.0*

on Wood:

95.4

97.0

96.6

95.0

95.2

94.9

94.7

93.7

93.0

92.6

on Plastic:

90.7

86.6

81.9

79.9

79.5

76.3

73.8

70.5

67.7

66.5

Table I: Coefficient of restitution as a function of impact speed, e(U)*100
n.b. The error in the values of e(U)x100 is less than 0.2 in nearly all cases.
* The impact speed was 6.45 m/s in these two cases.

Tests were made at ten speeds, six for the drop tests between ~2 and 6.5 m/s, and four for the pendulum tests between ~0.5 and 2 m/s, with an overlap at 2 m/s to check for consistency between the two modes of test. Each test for the final set of data consisted of ten drops, so that the data shown in table I and figure 3 are the result of about one thousand trials, and, taking into account the preliminary tests to establish the conditions for internal consistency, and the intercomparison of the different balls from the same maker, probably several times more. To illustrate the precision of the results, the ten drops of J on wood at U = 6.01 m/s had fall times, tf, in the range 15.41 - 15.47 ms, with mean 15.44 ± 0.01 ms (standard error of ten readings, shown hereafter as e.g. 15.44(l), where the figure in brackets is the uncertainty of the previous digit). The corresponding values of the speed ratio, r, were between 0.784 - 0.789, with a mean value of 0.788(l), giving e = 0.795(l) when corrected for extra acceleration using equation (3). At the lowest speed of the drop tests r = 0.733(l), needing a large correction to give e = 0.827(l), while at the same impact speed in the pendulum tests, r = 0.823(l), which there needed only a tiny correction to e = 0.824(l) from equation (4); the closeness of the calculated values of e from the two modes of measurement is reassuring.

graphs of coeffcient of restitution against speed
Fig. 3. Collected data for the variation of coefficient of restitution with impact speed, e(U). Symbols:- squares, Jaques ball; circles, Walker ball; triangles, Townsend ball. Colours:- red, impact on steel; blue, impact on wood-, black, impact on plastic. The green lines indicate the acceptance limits of the CA bounce test (see text).

The balls from each maker sometimes showed systematic differences, particularly for the J balls. For instance, a J ball gave values of e of 0.823(l) and 0.801(l) from pendulum tests at 2.04 m/s on closely controlled impact positions at opposite ends of a diameter, while tests at the ends of a diameter at right angles to this gave values of 0.828(1) and 0.813(1); a different J ball under the same conditions gave values of 0.833(1) and 0.814(1). The variation over a single J ball, and also over the four nominally similar J balls, were at a level of ±0.020 from their average, about twenty times greater than the uncertainty of the measurements. The W and T balls were more uniform. The two T balls gave the same e(U) values to ±0.002 without any need to control their impact positions, while the two W balls were only a little worse, at ±0.005. To avoid this variablility from confusing the data for the systematic dependence of e on impact speed e(U), especially for the J balls, all the results shown in table I and figure 3 were taken on a controlled impact position for a single ball of each type.

The value of e falls off appreciably if the impact position approaches the edges of the mallet face, though the effect did not become significant until it was more than 8 mm off centre, which was much greater than the spread of impact positions. It was also confirmed that altering the clamping pressure on the wood and plastic mallet heads over a wide range of tightness made no difference to the values of e(U).

4. Results and Discussion

4.1 General features: The results show a wide range of behaviour for the dependence of e on impact speed. In most cases the systematic variation of e(U) is smooth at the level of precision of these measurements (i.e. ± 0.001, which corresponds to a variation of about 0.1" in the CA bounce test), though the scatter is worse for the W ball, which also has an apparent discontinuity between the pendulum and the drop data for impact on steel. No reason is known for these small discrepancies for the W ball (though the results of Le Maitre, [4], suggest that they could be due to the milling pattern which was not controlled between the drop and the pendulum tests for the W ball, or perhaps due to changes of laboratory temperature), however, since its behaviour for impact with both steel and wood indicates that it has an essentially constant value of e on these surfaces, the reason for the discrepancies was not pursued.

Seven of the nine combinations of ball and surface in figure 3 show a variation of e(U), with only the J ball lying within the current CA acceptance limits for the whole range of speeds and surfaces. The results for J on wood and on plastic are clearly different, which raises doubts about the application of Law 3e(3), as mentioned previously, but both are fairly close to the results on steel, the standard CA test surface, and it is unlikely that the nature of the mallet face would make much difference when playing with J balls. However, if as has been suggested, [4], the CA acceptance limits were to be tightened (e.g. limits of 31" to 37", i.e. e between 0.72 and 0.79, as shown by the dashed green lines in figure 3) to be more appropriate for a modern sport, then differences of the sizes measured here would probably need to be taken seriously.

Both T and W balls had high values of e on wood and on steel, particularly the T ball - which was not far from ideally elastic (which many players found bad for some croquet strokes), and neither showed much dependence of e on impact speed, U, particularly the W ball. On plastic, however, both showed a marked decrease of e with increase of U, as is discussed further in section 5.3.

4.2 The e(U) characteristics: The coefficient of restitution, e, is a measure of the loss of kinetic energy as a result of impact, the fractional loss being equal to (1 - e2). This energy loss depends on the anelastic response functions of the two materials involved. Zener, [5], gives an introduction to the physical ideas involved, and [3] gives a more detailed discussion of them in the context of croquet, including some of the issues mentioned later in this section. The parameters involved for each material are its value of Young's modulus (i.e. its 'stiffness'), and most importantly, the hysteresis ('dragging behind') between the applied stress during impact and the mechanical response inside the material; it is this hysteresis which gives rise to energy loss, and an important feature of the hysteresis is that for some materials it can become large over the time scale of the applied stress, which is ~l ms for the duration of impact for a croquet ball.

The advantage of steel for the CA bounce test is that it is so much stiffer than any croquet ball dropped onto it that it suffers negligible deformation during impact, and thus contributes nothing to the total energy loss during collision: the measured behaviour of e(U) thus represents the intrinsic behaviour of the ball. Wood and plastic (at least that used here) are both much softer than steel, and about as soft as the balls dropped onto them, so that in these cases both of the materials involved in the collision suffer deformation, and both may contribute to the total hysteresis loss.

From the curves in figure 3 it is apparent that the balls dropped on wood give e(U) values which are qualitatively similar to those for drops on steel, and also usually rather higher in value, indicating smaller energy loss. It is at first sight surprising that the energy loss could be less than for drops onto steel (i.e. the intrinsic loss of the ball), but the explanation of this apparent anomaly is that the factor which determines the energy loss is not the impact speed, but the amount of deformation of the two materials involved in the collision, [3], and for the ball striking wood this is considerably less than it is for impact with steel. Since the total energy loss has decreased, it is evident that the energy loss in wood must be rather small, though the results for the T ball on wood and on steel at low speed show that it is not negligible. The same considerations show that although e(U) for bounce on steel represents the intrinsic energy loss of the ball, nevertheless, the form of e(U) will probably be different in ball-ball collisions since the balls will be deformed by a different amount from impact with steel; to find e(U) for ball-ball impact could probably best be done by pendulum type ball-on-ball tests.

The results for impact with plastic show enhanced energy loss (rising to more than 50%), despite the fact that the deformation of the ball is much less than on impact with steel, and probably less than on impact with wood. This necessarily implies that the energy loss in the plastic is large, and considerably more important than the loss in the ball. This is of course only a qualitative explanation of the differences between the curves, but a quantitative explanation of the form of e(U) would require a lot more information about the anelastic behaviour of the materials involved (see e.g. [3]), and would not be particularly useful.

5. General Discussion and Conclusions

5.1: The present acceptance limits of the CA bounce test were set very wide (partly to encourage the development of alternative balls), but tighter limits on bounce are used in ball selection at the highest levels of the game, cf. [4], and if these were to become CA norms, in keeping with the development of the game as a precision sport, then it is clear that a more convenient method of ball-testing would be desirable.

The apparatus described here appears complicated compared with that of the simple drop test, however, it has the advantage that once set up it enables a large number of measurements to be made easily and rapidly, and for a given arrangement, where the dimensions involved in the correction factors of equations (3) - (5) are known, the electronics can be configured to give a direct readout of the value of e with an uncertainty of about ±0.15%, which is equivalent to an uncertainty of about ±0.1" in the rebound height of the ordinary drop test (n.b. the initial drop height is not critical for the system described here, since the timing system measures the ratio of speeds before and after impact, whereas for the simple drop technique it must be controlled to better than the uncertainty in the rebound height). Accuracy at this level would allow the study of a number of problematic issues such as; the effects of temperature, of milling, of ball uniformity, and of wear.

5.2: The results shown here are only of qualitative significance, since of the six materials used in the tests, only the steel has well known physical properties. The other materials were typical of playing use, but were not characterised in sufficient detail to allow any quantitative conclusions to be drawn, and in particular, the makes of balls tested are now obsolete. It would be interesting to have measurements of e(U) for current tournament balls, to see if they differ significantly from the "Eclipse" balls which they have replaced, but they had not been developed at the time of this work.

5.3: The results show clearly that the details of impact in croquet strokes depends on the materials of both ball and mallet. The ball is subject to some control by the drop-test Law, as discussed above, while the mallet has only the loose restriction on its construction given by the wording of Law 3e(3) (see Introduction). It is evident that mallet heads can behave considerably differently from wood, and the results suggest that appropriate choice of materials would enable their response to be 'tuned' to a player's preference, but whether this would confer any "playing advantage" remains problematic. The only material which is banned for use on mallet faces is rubber, but in view of the huge variety of modern polymeric materials, it is not easy to see that there is a clear distinction between (banned) rubber and (acceptable) other soft polymers, and it would seem not unreasonable to allow mallet faces to be made of any material that a player prefers, subject only to causing no damage when striking the ball.

5.4: The behaviour of e(U) is also important in two-ball strokes. These have been analysed by Calladine and Heyman, [6], who consider first elastic behaviour, and then extend this to include lossy behaviour with a constant value of e for ball-on-ball collisions. They choose the value e = 0.75, which figure 3 shows to be a realistic average value, and calculate its effect on straight and split croquet strokes. The results give an instructive insight into the different strokes used in the game, though their treatment is necessarily approximate, since a full theory involves solution of a difficult three-body problem (mallet and two balls, in the presence of ground-ball friction and ball-ball friction, and with different coefficients of restitution for the two contacting surfaces, ball-mallet and ball-ball).

Modelling a real impact using the coefficient of restitution is only an approximation to the detailed mechanics of the collision. Similarly, "pull" during split croquet strokes, particularly important when peeling, is also something where the detailed nature of the surface contact is important (cf. Daish [1], pp 156 - 161).

Appendix on Lawn Speeds

A1. Experiment: "The speed of a lawn" is commonly found by striking a ball so that it just travels the length of the lawn, and measuring the time it takes; this time is longer for a faster lawn, which can be understood from Newton's laws with constant deceleration. However, the frictional force on a moving ball depends on the length, dampness and quality of the grass, and on the smoothness of the underlying surface, so that whether or not it is reasonable to assume constant deceleration is something which needs to be tested by experiment. By varying the distance of travel, l, and measuring the time taken, t, one can study the relation between them, t(l), to test whether it is compatible with constant deceleration. For translational motion of a mass with initial speed, v, subject to deceleration, f, the relations between the variables are given by,

l = ft2 / 2;     v2 = 2fl;       l = vt / 2.
(A1 a, b, c)

Rolling adds extra terms to the equation of motion, but these are of similar form to those above and equations (A1) are usually adequate to describe the motion of a ball over a lawn, as discussed further below.

lawn speed graph
Fig: A1. Test of lawn speed; time taken, t seconds, to come to rest at a distance 1 metres (cf.eqation (A1a)).

Equation A1.a was tested for values of 1 between 3 and 15 m. Reasonably consistent results were obtained for repeated measurements over the same area of lawn, and typical results are shown in figure A1. The results could all be fitted by a linear relation between l and t2, with values of deceleration, f, between 0.6 - 1.2 m/s2 for different conditions of the lawns. These values correspond to times of 7 - 10 seconds for the normal "speed test", and evidently show that the early season lawns were rather slow. The initial speed required for a ball to travel different distances can be calculated from equation A1.b when f is known, and range from 12.5 m/s for a full lawn shot on a slow lawn (f = 2.5 m/s2), to 0.25 m/s for a 10 cm shot on a fast lawn (f = 0.3 m/s2). For two-ball strokes, and taking into account the energy loss at impact, (1 - e2), the speeds necessary could be much higher.

A2. Rotational motion: A more complete treatment of the motion of a ball, including rotation, has been given by e.g. Daish [1]. Spin is important in snooker as it can be imposed by cue action, but it is much less so in croquet since the size of a mallet face causes the impulse of the mallet stroke to pass through the centre of mass of the ball. The striker's ball always starts motion without spin, unless there is some fictional couple acting on the ball, as e.g. in the downward stroke of the jump shot, or as in the "pull" caused by contact with another ball in a split croquet stroke. The initial motion of the ball is to slide until the frictional couple from the grass has accelerated the rotation of the ball until there is no relative motion between the point of contact of the ball and the grass: at this stage, the linear velocity of the centre of mass of the ball, v, is equal to the velocity of the surface of the ball about its centre of mass, wR, where w is the angular velocity and R is the radius of the ball.

The equation for rectilinear motion of the centre of mass is,

m (dv/dt) = F,
(A2)

where the force, F = -µmg, and µ is the coefficient of friction, here assumed to be constant. The equation for the angular velocity, w, about an axis through the centre of mass is,

I (dw/dt) = G,
(A3)

where I is the moment of inertia of the ball, 2mR2/5, and the torque, G, is the product of the same force (since the only force acting is at the point of contact between the ball and the grass) and the radius of the ball, i.e. G = µmgR. Integrating equations (A2) and (A3), and using the initial conditions that v = vi, and w = 0 at t = 0, gives

v = vi - µgt,
(A4)

and

wR = (5/2)µgt.
(A5)

Sliding stops when v = wR, i.e. after a time ts,

ts = (2/7)vi/µg,
(A6)

at which time v has fallen to a characteristic value,

vs = (5/7)vi.
(A7)

For a constant value of µ, (A4) predicts that the velocity will fall at a uniform rate until the ball stops, as indicated by the black line in figure A2; this is exactly what would be expected from equations (Al) and is some justification for their use. However, there is good evidence that friction is higher during sliding than during rolling, [7], and if both regimes have constant coefficients of friction, µs and µr, respectively, then v(t) follows two linear regimes with a change of slope from µsg to µrg at the critical velocity given by (A7), as indicated by the green line in figure A2.

If the friction varies with the ball position, µ(l) say, as seems highly likely for motion on a typical lawn, then equations (A2) and (A3) cannot be integrated easily, and the form of v(t) is more complicated. One possible behaviour is indicated schematically by the blue line, which represents an 'uppish' stroke where the ball leaves the ground (and moves through the air with negligible energy loss and thus constant velocity), makes contact briefly with the ground and starts to rotate, bounces again, and finally behaves as for the green line. The behaviour for a stroke with downward pressure (though not sufficient to cause the ball to jump) is represented by the red line in figure A2, where the coefficient of friction falls from a high initial value due to a large relative speed between the surface of the ball and the ground, to the value for the rolling regime when the relative speed becomes zero. Finally, the brown line in figure A2 represents a jump shot; this stroke imparts a lot of spin to the ball because the large downward impulse (which acts so quickly that it cannot be separated from the process of the stroke) greatly increases the frictional couple acting on the ball. When the ball touches the ground, the spin may well be greater than that needed to sustain the rolling regime, and the friction will now act so as to increase the value of v, while decreasing that of w.

sliding to rolling motion against friction
Fig. A2: Schematic diagram for v(t) for the transition from sliding to rolling motion for various forms of friction; see text

A3. Energy considerations: The work done against friction can be calculated by integrating the product of force times distance moved along the v(t) paths illustrated in figure A2: the force acting at any instant is just m(dv/dt), since the acceleration is just the gradient of v(t), while the distance moved in the increment of time dt is just v(dt). Thus, integrating the product mv(dv) along the path between any two points gives the difference of kinetic energy, mv2/2, between them, and this must be independent of the particular path taken. The initial kinetic energy of the ball is mvi2/2, while at the onset of rolling the total kinetic energy (translational and rotational) has fallen to 5mvi2/14, a kinetic energy loss of mvi2/7 due to friction. Calculating the work done against friction due to the centre of mass motion shows it to be 12mvi2/49, i.e. greater then the loss of kinetic energy, but the difference arises because the friction also does work which can be calculated to be 5mi2/49 to set the ball into rotation, and, for all the curves shown in figure A2, this exactly accounts for the difference. The linear velocity at the onset of rolling is a characteristic fixed fraction of the initial velocity, independent of the value and the detailed behaviour of the friction. The time to the onset of rolling, however, may vary considerably, cf. figure A2.

A4. Lawn speed: The black line in figure A2 shows that it is only when µ is constant that v(t) follows equation (A4); sliding usually occurs, and this affects the usual practical measurement of lawn speed since it is only when v < 5vi/7 that equations (A1) apply. The total distance travelled is given by integrating v(t) with respect to time, i.e. by calculating the total area under the curves of figure A2. For the black line this is just the area of the triangle, vit/2 (cf. equation A1l.c), but for the red and the green lines of figure A2 (i.e. those produced by fairly smooth strokes), the area of the triangle formed by linearly extrapolating the rolling regime of v(t) to t = 0 (cf. the dashed extrapolation of the red line in figure A2), does not differ from the area under the true path by more than a few percent; for instance, Hall [7], who discusses the onset of rolling in a way similar to that given here, considers the case where µs = 0.48 and µr = 0.065, and shows that the difference between the two areas is only 1.6%. Provided that the strokes are smooth, lawn speed tests which rely on measurements of the time, t, to travel a total distance, l, can give values of frictional deceleration with errors of only a few percent, probably less than variations in µ over different parts of a lawn (which would give an extra 'waviness' to the lines in figure A2). These considerations give the theoretical justification for analysing the results of figure Al ignoring rotation. The agreement with equation (A1a) is reasonably good, though there are some signs that the data at higher values of l are systematically high, which could indicate that sliding was occurring in the stronger strokes, despite attempts to reduce it, cf. also A5 below.

A5. 'Lawn friction', F. An improved procedure for measuring lawn speed would be to measure l and t over the later parts of a drive, e.g. by playing a half-lawn drive from south boundary and measuring t not from the moment of impact, but from the time the ball passes hoop one, say, until the time it stops. The average of about ten such measurements of the quantity 2l/t2 in different directions and over different parts of the lawn would give a good value for the average deceleration of a lawn, f, while the standard deviation would also give a good impression of its uniformity, since most of the deviation is likely to come from real variations in the lawn, not from random errors of measurement, which are likely to be proportional to 0.2/t (where t is measured in seconds).

It is physically more instructive to divide the value of the deceleration, f, by the gravitational acceleration, g, to obtain a dimensionless ratio analogous to a coefficient of friction, a 'lawn friction', F, say. Expressed in this way, a direct comparison can be made with a component of gravitational acceleration due to lawn slope. For instance, in indoor tests of a ball rolling on vinyl tiles, F = 0.0074 ± 0.0014, and on haircord carpet, F = 0.018 ± 0.002, where in both cases the spread of the measurements was a gravitational effect due to uneven flooring at the level of ~0.002, i.e. ~2 mm over a distance of a metre.

The values of F found for croquet lawns are much larger than these (though perhaps not for indoor courts), but on the other hand the variations of slope on outdoor courts are also often much larger, particularly at their edges. For instance, local slopes of 1" in 3 yards were typical for the lawn which gave the data in figure A1 above; they contributed ~ ± 0.01 to its early-season value of F = 0.074 ± 0.007, and account for much of the measured standard deviation. The effect of slopes on the course of a ball is fairly small under slow conditions, but in high-summer when the lawns become fast and F can fall to ~0.03, the slopes can be expected (and be easily seen) to have a serious effect on the course of a ball.

Acknowledgements

These experiments began as final year undergraduate projects. I am grateful to several students for their enthusiastic help, in particular to D. G. McDowall for developing the timer circuitry.

References

[1]. C. B. Daish, The Physics of Ball Games, E. U. Press (1972)

[2]. V. K. Gupta et al., Am. J. Phys, 54 (1986) 619 - 622

[3]. D. Gugan, Am. J. Phys, 68 (2000) 920 - 924, www.oxfordcroquet.com/tech/gugan/index.asp

[4]. R. Le Maitre, Croquet Gazette, 228 (1993) 8 - 9, www.oxfordcroquet.com/tech/rml/2/index.asp; see also T. Haste, ibid, 227 (1993) 18

[5]. C. M. Zener, Elasticity and Anelasticity of Metals, U. of Chicago Press (1948)

[6]. C. R. Calladine and J. Heyman, Engineering, 29th June (1962) 861-863, www.oxfordcroquet.com/tech/calladine/index.asp

[7]. S. Hall, www.oxfordcroquet.com/tech/hall/index.asp

Author: Don Gugan
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