Technical
The Physics of Croquet Strokes: Analysis of the CA highspeed DVD
Don Gugan, Bristol 1. IntroductionWhat happens during croquet strokes is still not completely clear. A theoretical framework was given many years ago by Calladine & Heyman [1], but they simplified their analysis by the almost complete omission of the effects of friction and spin. Several experiments have been reported since then which illustrate the complications that these produce: using video recording [2, 3], electrical timing [4, 5, 6], carbonpaper impressions [2, 7, 8] and friction measurements [4], but up till now they have not allowed a quantitative study of the crucial millisecond or so around the time of impact. This has changed with the highspeed DVD of croquet strokes sponsored by the Croquet Association and now made generally available (see Dawson, The Croquet Gazette No. 312, p9) [also online]. The reason for making the DVD was to clarify certain problematic matters of the Laws and their interpretation by referees, such as doubletaps, prolonged contacts, etc., where high timeresolution was necessary, but a great deal of potential interest about the physics of croquet strokes is also contained the images, some of which is discussed here. For those who have not already seen the DVD, some description of its contents is necessary. The disc contains three folders, frequently referred to by name in what follows: (i), TEXT, a file which contains a summary of the various trials made, and the preliminary conclusions; (ii), HSAVI, a folder which contains about 70 closeup, highspeed video files in avi format, of which 39 are suitable for detailed analysis and allow images taken at a rate of 8000 per second (125 microseconds/frame) to be viewed at full size on a computer screen and stepped through one by one; and (iii), STDMPEG, a standard speed video which shows the overall lawnview of each of the highspeed files. 2. Remarks on the DVD ImagesThe HSAVI images show the motion in the xyplane as a result of straight strokes along the xaxis, viewed from a few feet perpendicular to the point of impact, and slightly above the centre of the ball; the images are about 200 x 40 millimetre (mm) when expanded to lifesize, cf. figure 1(a). The balls had meridianal reference circles marked on them to indicate their rotation about the zaxis, and in principle it should be possible to measure the x and y positions, and the angles in the xy plane for ball(s) and mallet to about ± 0.2 mm and about ± 0.2 degrees of arc. In practice however, the accuracy is reduced because the images show less than half the height of the ball, because the mallet had no clear reference marks, and also because of parallax, of indifferent focus, and of shadow over crucial parts of the image: the uncertainties of measurement prove to be several times greater than might be hoped. It proved helpful to make a transparent, ballsize mask of a circle with two perpendicular sets of ellipses about its centre. This could be superimposed on the visible edges of the image of the ball on the screen, used to scale it to lifesize, and rotated to conform to the positions of the meridianal circles, as shown in figure 1(b). The xdisplacements are easily found with respect to an edge of the image frame, while the angular rotations can be measured using the reference circles marked on the balls, though not easily as the images show them obliquely, causing the reference lines to appear as arcs of an offaxis ellipse. The ydisplacements of the ball (cf. the downward movement of the ball shown in the two parts of figure 1(b)) can best be found from the change in length of the chord intercepted by the upper edge of the image frame, using the result illustrated in figure 1(c) that for a circle of radius r, the chord length, l, and distance along its perpendicular diameter, y, are related by (l/2)^{2} = y(2r  y). This equation can be solved to give the vertical displacement of the ball from its original level: though not striking on the images, this is sometimes very important. Measurements on the images were much assisted by laying the computer screen flat. A difficulty of analysis is that no information is provided about the consistency of the strokes. The STDMPEG video makes it clear that some of the shots were repeated, and also that at least two players were used to make them; however, while the same type of stroke is repeated for different strengths, no ‘identical’ strokes were attempted, in order to give some idea of the inherent variability. The repeated strokes show a general similarity on the STDMPEG video, but significant differences in the HSAVI closeups – but how important these are is impossible to assess. For the original purpose of the DVD there is advantage in having a player rather than a mechanism making the strokes, but the absence of precise and repeatable control limits the conclusions which can be drawn. A further difficulty is the lack of sufficient background information. What one sees is a mallet striking ball(s), but it is well known that the type of mallet, of ball, and the quality of the ground all affect the outcome of a stroke. The most important unknown is the nature of the surface at the position of the stroke; it is the initial reference level for each stroke and is responsible for much of the behaviour, but it is not shown on any of the videos, and it is often impossible even to know if the balls remain in contact with it. In any event, it is hard to believe that a grass lawn can be the welldefined, smooth and level surface needed for mathematical analysis, or that it would not be affected by the succession of strokes made on it. Again, for the original purpose of the DVD these unknowns are immaterial, but they are a limitation on serious analysis. 3. An Overview of the DataThe 39 HSAVI videos contain three different series of shots: series A, with 8 singleball strokes of increasing strengths, – 4 drives and 4 rolls; series B, with 16 straight scatter shots at a variety of strengths and ball separations (though the 5 strongest strokes prove to have nothing of interest here); and series C, with 15 twoball croquet strokes, – 3 drives, 2 halfrolls, 4 fullrolls, 3 passrolls and 3 stopshots. The basic information measured from the videos for each of these strokes is contained in the tables 1  6, viz. the horizontal component of ball and mallet speeds, the angle(s) of the mallet stroke, the contact times, and the rotational speed of the balls measured over the same period as the linear speeds; a few of these are also reported in TEXT on the DVD and usually differ only by small amounts due to the joint uncertainties of measurement – though in a few cases there are significant differences which are discussed later. The tables also contain values for the total travel distances of the balls taken from TEXT. For the rolls the vertical motion is also important, and basic information about this is given in Table 4(b), and illustrated in figures 3(a) and 3(b). The lower part of each of the tables contains quantities of interest calculated from the basic measurements [shaded yellow]. The video strips typically run for about 200 frames (i.e. 25 milliseconds (ms)) and show travel distances of about 50 mm for the balls and the mallet – though often less when the speeds are low. Measurements of position and angle were plotted against frame number to find the speeds close to the time of impact, and to try to detect any nonlinear changes; however, as the measured path was always only a very small part of the total travel distance, the few nonlinearities observed were usually too small to be relied upon for detailed analysis. The values of speed in the tables are thus averages taken a few milliseconds after impact, and over times and distances of only a few ms and mm respectively: because of this, their precision is limited to a few percent, though it is convenient to retain higher precision in the tabulated values. It is important to realise that an uncertainty level of several percent is probably the best that can be extracted from the video data. The most important difference between the different strokes is whether the striker’s ball rises immediately from the ground at the moment of impact, or is pressed downwards. For the first, what may be called “uppish” strokes, there is no reaction with the ground, and spin is imparted to the ball only from the friction between the ball and the mallet face, and a good theory to describe what happens is given in Appendix B. For strokes with initial downwards movement there is inevitably a further (and usually dominant) force from ground friction; the situation is far more complicated, and no general theory appears to be possible because the region of contact is deformable, and the precise nature and the relative sizes of the forces from ground and from mallet friction are unknown. Further discussion is given in Appendix C. The results presented here start with the simplest collisions, the ball on ball collisions involved in the series B scatter shots (section 4); next the singleball drives from series A (section 5), then singleball stopshots from series B (section 6): all of these ‘uppish’ strokes are then followed by the the singleball rolls from series A (section 7). The 2ball croquet strokes without appreciable roll, and finally, the twoball roll strokes of various types, all from series C, are discussed in sections 9 and 10 respectively. There are in fact further highspeed videos on the DVD, but they do not deal with collisions and are not analysed here. 4. Ball on Ball Collision (HSAVI files, series B)These videos all show straight scatter shots, often problematical because of the potential for double taps when the scattered ball is close. The fault is least likely when the striker’s ball is hit with a stopshot action to avoid putting roll on it, so that it skids over the grass without spin and then collides headon with the scattered ball. Under these conditions the collision between the balls conforms closely to the Newtonian theory of colliding pointmasses, as outlined in Appendix A but with the mass ratio, k, equal to unity. Thus, when the striker’s ball, ball 1, moving with speed v_{1} strikes an identical target ball at rest, ball 2, the final speed of the target ball is
where e is the coefficient of restitution (abbreviated to CoR hereafter), a measure of the energy loss on collision, and required by CA regulations to lie between 0.71 and 0.87 – and in practice usually much nearer the lower limit. The coefficient e is important for all the collisions shown on the DVD, but is not specified in the supplementary information in TEXT: these ball on ball collisions give the best measure of its value, which is used extensively in what follows.
In these experiments the scattered ball was put on the xaxis a short distance from ball 1, and for many distances a fault inevitably occurred. However, several of the strokes with the larger separations could be analysed despite any doubletap because the fault occurred after the ball on ball collision. Table 1 contains results for ball separations of 33 and 13 mm, arranged by increasing strength of shot; many of them were classified in TEXT as double taps, indicated in the Table by halfshading, and examination of stroke B7MS showed that although not classified as faulty, it too was a clear double tap. The values in rows 1  3 of the Table are taken from TEXT; rows 4  7 contain the basic information about the collision, and rows 8 and 9 give the results of various calculations discussed below. The stroke used gave a small amount of yelevation to the striker’s ball, but not enough to affect the analysis of the collision as in the xdirection only. Likewise, the balls have a small amount of spin which could in principle affect the results; however, the spin of the striker’s ball was very small (cf. section 6 below), and while reliable measurements of spin of the scattered ball were not possible, the evidence suggests that it must have been close to zero, so that the impact was essentially without rotation and equation (1) thus an excellent approximation (cf. also section 5.3). The ratio of speeds of the two balls as a result of impact, v_{2}/v_{1}, is tabulated in row 8 of the Table, and has a mean value of 0.86 ± 0.01, where the error limits are the standard error of the mean (sem) arising from the random errors of measurement. This yields e = 0.72 ± 0.02, a value which cannot be independently confirmed, but is certainly much as would be expected for approved croquet balls. The ball on ball contact times, T_{b}, (row 5) show no significant variation with ball speed, and cluster around 0.75 ms (i.e. 6 video frames), close to those for mallet on ball contact times which are discussed later (section 9.2). 4.1. ‘Clean’ Scatter ShotsFor an ideal straight scatter shot with perfectly elastic balls, i.e. e = 1, ball 1 would stop dead; in reality it retains a small speed after collision, u_{1} say, and equations (A3, 4) of Appendix A show that the speed ratio of the two balls after collision is
The square of this ratio is the ratio of the kinetic energies of the two balls, and thus, as discussed in section 5.4 in connection with lawn friction, the ratio of the distances that the balls are expected to travel over a uniform lawn before friction brings them to a stop, L_{2}/L_{1}. Substituting the mean value just obtained for e gives a predicted value of 38 ± 7 for this ratio: by comparison, the CA limits on e, 0.71 to 0.87, lead to limits on the distance ratio of 35 to 207. The CA upper limit on e could well be considered unduly elastic. The three clean scatter shots in Table 1 have values of L_{2}/L_{1} which agree well with the predicted value; the value for B5MS being even greater at 49, possibly because this stroke was one of the very few played by a different player, using a more extreme type of stopshot action. In the faults on the other hand, the doubletaps always have the effect of giving the striker’s ball extra speed, of increasing its travel distance L_{1 }, and thus reducing the distance ratio L_{2}/L_{1} below its value in a clean stroke. The effect is evident in row 3 for the halfshaded entries, with about a tenfold reduction of the ratio for a clean stroke. The intermediate value of L_{2}/L_{1} for stroke B7MS, however, needs clarification: the second contact in this case was fairly gentle (and overlooked in the initial analysis, and wrongly classified in TEXT), but the extra impetus given to the striker’s ball was sufficient to reduce L_{2}/L_{1} to about a quarter of its proper value. When refereeing a close, straight scattershot, the best test is this distance ratio, and the physics of the situation requires that for a clean shot the ratio must be greater than a lower limit, here calculated to be 38 – though in view of the difficulties of adjudication, a lower value of about 25 might be chosen as a working limit for the purposes of the Laws. It is clear that the erroneous classification of B7MS would suggest a limit far too low. 5. Singleball DrivesThe results for four drives of increasing strength (HSAVI files A1D  A4D) are given in Table 2. Most of the speeds given in rows 7 and 9 of the Table have only small differences from those in TEXT, but the difference in ball speed v for A4D (10.03 m/s here, 8.62 m/s in TEXT) is significant since the TEXT value (and the graph in the Gazette article) leads to inconsistent and physically implausible behaviour of the ratio v/V, cf. the discussion in section 5.2 below.
The four drives were made with a pendulumlike swing with a nearly horizontal mallet head at impact, and the geometry of the collision and the forces acting during impact are shown in figure 2(a). The mallet strikes the ball at point C at the mallet angle, α, the same as the forward angle of the mallet shaft, and which when positive tends to put roll on the ball; however, the direction along which the mallet moves, the stroke angle, β, is negative for all the drives and has the opposite effect – though which dominates cannot always be easily predicted. The angle α is easily found from the video strips, but β is more difficult to measure as the mallet had no reference marks on it; its line of motion just before impact can only be estimated indirectly, but the results indicate that the stroke becomes nearer to horizontal as the strength increases. Motion in the ydirection can also be important: the negative value of β used in the drives here results in uppish strokes and the ball leaving the ground, as can be seen on the STDMPEG video for the stronger strokes, but the HSAVI closeups have poor yresolution and no upward rise can be detected there. However, the angles involved are small, and the impact can still be analysed to sufficient accuracy considering only the horizontal motion. 5.1. Horizontal Motion.The collision can be modelled using the equations from Appendix A;
and
where U and V are the initial and final values of mallet speed, and v is the final ball speed; k is the ratio of mallet/ball mass, and e the CoR, which one expects to be closely the same as for the ball alone since few mallets contribute to the energy loss. Equations (A3, 4) contain an implicit assumption that there is no spin of the ball, which the data in rows 6 and 10 of Table 2 show is not the case; however, as discussed in section 5.3 below, it is small enough to ignore for strokes A2D  A4D, while the case of A1D is there discussed further. Equations (A3, 4) show that the speed ratios (v/U) and (V/U) depend only on the terms k and e, and should therefor be constants: rows 11 and 12 of Table 2 confirm this within the uncertainties of measurement. The value of e can be found directly from the measurements as it is by definition the ratio of the relative speeds of separation after collision to that of approach, i.e. (v  V)/U, as in row 13 of Table 2; the values vary by rather more than one might expect, though in no systematic way, and their average value e = 0.74 ± 0.04 appears to be the best interpretation of the data, and in good agreement with the value of 0.72 ± 0.02 found in section 4. Equations (A3, 4) depend otherwise only on the mass ratio, k, and assuming for e the previous best value of 0.72, they lead to a values for k of about 2.5, – though no firm comparison can be made with expectation since the necessary information is not to hand. No doubt the mass of the mallet used could be obtained, but it is important to remember that the crucial factor in the physics is the effective mass of the body involved in the collision; this will certainly be less than the total mass of the mallet because its centre of mass is not at the point of impact – and it will also depend on other factors including bodily motion, flexure of the shaft, and firmness of grip. 5.2. Contact time, Τ, and Distance, XEven at 8000 frames per second the duration of contact is hard to measure because of lack of a clear view at the points of contact and separation, cf. figure 1(b). The initial contact can be identified by the first slight movement of the ball, but its end is often uncertain to a frame or two – though a helpful secondary indicator is that in the middle of the contact period the mallet and ball have no relative motion. The compression of the ball can be seen on superimposing the circular mask over the image, and is easily seen to be about 3 mm for the drive A4D, however, the resolution is not sufficient for compression to be studied in detail. Hertz’s theory of impact, cf. [6], predicts that the shorter contact times occur for the stronger strokes, but the effect is weak (i.e. constancy of the product Τv^{0.2}, a correction factor of only 1.6 for the tenfold range of speeds used here), and while the corrected values of 0.99, 0.99, 0.91, and 0.99, are indeed constant as predicted, the uncertainty in the values of T makes it not worth while to make the correction in values of T reported hereafter. The contact distance, X, in row 5 of the Table, is the horizontal movement of the ball during the duration of contact. The mean speed of the ball during contact is v/2, so that the product X/vΤ should be equal to 0.5, and the calculated values of 1.0, 0.47, 0.75 and 0.50 agree with this within the uncertainties of measurement. The size of X is significant in considering the possibility of making a crush fault, but as these measurements confirm, even for strong strokes the contact distance is very small, and the doubletap is always the more likely fault. 5.3. Spin During DrivesSpin is acquired by a croquet ball as a result of frictional forces from the ground, as in roll strokes (and of course in the transition from sliding to rolling), and from the mallet face. Figure 2(a) illustrates the forces during impact, where the strokes are controlled by the mallet angle, α, and the stroke angle, β, which usually differ considerably: the mathematical technicalities are discussed in Appendices B and C for mallet and ground friction respectively. The values of the angular rotation of the striker’s ball after impact can be converted to values of surface speed of the ball, v* say, and this is tabulated in row 6 of Table 2, but the most convenient way of considering roll is to compare it with the linear speed of the centre of the ball by defining the fraction of pure roll, i.e. Ω = v*/v. Ω is equal to unity for a ball rolling over a surface without sliding, but immediately after a croquet stroke it will usually be different: close to zero (or even negative) for a stop shot, close to unity for a roll, and in between for drives. The values of Ω for the drives are shown in row 10 of Table 2: for the stronger strokes they are rather constant at about 10%, but for A1D, though apparently played with much the same mallet action, notably larger at 33% – though without information about the natural variability of the strokes there must be some doubt about how significant this difference is. The kinetic energy of the roll motion is energy which has been lost from the final kinetic energy of linear motion of the ball, and it therefor causes an apparent reduction of its CoR by an amount (cf. Appendix A),
The kinetic energy of pure roll is mv^{2}/5, i.e. 40% of the translational kinetic energy, so for the strokes A2D  A4D where Ω ≈ 0.1, only about 0.4% of the energy of the ball is due to rotation – negligible at the present level of accuracy. Even for A1D with Ω = 0.33, only about 5% of the energy available for translation has been lost in producing roll, which implies that the apparent value of e in row 13 of the Table should be 0.76 (i.e. equivalent to correcting the ball speed in row 9 from 0.97 to 0.99m/s): this is not a negligible effect, but it is near the limit of the accuracy of measurement and not large enough to affect any of the previous discussion. How the initial spin arises, and why the apparently similar type of stroke used for all four drives causes one to respond very differently can depend only on the two controllable variables, the mallet angle, α, and the stroke angle, β, both of which certainly vary during the series of drives, despite their superficial similarity. All the drives measured here were played with mallet and stroke angles expected to produce uppish strokes, and producing spin only from mallet friction (cf. Appendix B) given by
The values of Ω calculated from (B5) for the drives A1D  A4D are 11, 9, 14 and 10 % respectively: for the last three, stronger shots, the agreement with the measured values is excellent, fully in agreement with the ball being airborne and not experiencing ground friction, however, the anomalous value for A1D implies that some other force is contributing to the roll here. The motion of A1D as a result of the mallet stroke can be calculated to be a jump of length 40 mm, and maximum height 2 mm after a time of 20 ms, but its value of Ω was probably established within about 5 ms of impact, during which time the ball had moved horizontally by about 5 mm, but only 1 mm vertically. The significance of this perhaps lies in the nature of the contact between ball and ground: it is not an ideal sphere resting on a rigid mathematical plane as in the model of figure 2(a), but a ball sunk in a lawn covered with a layer of grass several mm in depth. If one assumes that in the stroke A1D the ball slid over the grass at a speed of 1m/s for 5 mm, until its upward component of velocity lifted it clear of the grass, then equating the impulsive torque µmgr (assuming that µ for sliding on grass is 0.5, [4]) over this time to the change of angular momentum gives a contribution to Ω of only about 5%, insufficient to explain the discrepancy. A further possibility is that some element of ground friction in the initial stroke could be involved here, when equation (C2) would be applicable, cf. Appendix C; if this were to be the case, then inserting the values of α and β in (C2) would give a possible value of Ω = 49% – but there is evidently an anomaly which needs further work for its clarification. 5.4. Lawn FrictionThe initial kinetic energy of the ball is dissipated by friction as it rolls across the grass. Using the usual expression for frictional force acting on a body of mass m with an assumed constant coefficient of lawn friction µ, i.e. force = µmg, and equating the initial kinetic energy mv^{2}/2 to the work done against this force over the distance travelled, L, we obtain
which is the basis for the usual methods of estimating lawn friction. However, it takes no account of the transition from sliding to rolling motion which must occur, cf. [4, 10]; the necessary correction to take account of spin is discussed in Appendix D, and leads to the result that equation (3) remains applicable, provided that v is replaced by v_{r} from equation (D1), where v_{o} and Ω_{o} are corresponding values of speed and rolling fraction
(n.b. if Ω_{o} = 1, equation (3) takes its original form, as to be expected). Values of µ obtained using the corrected form of equation (3) are shown in row 14 of Table 2: they vary by more than the expected errors of measurement, but the mean value of 0.05 ± 0.01 corresponds to times of 11.5 ± 1 seconds for the normal lawn speed test, typical of a good quality lawn. In fact these values of µ, like those from all the tests analysed here, show a systematic increase with increasing ball speed, and one may surmise that wear had flattened the grass in the immediate test area, while much less had occurred for the larger distances covered by the stronger strokes: since this has implications for the distance ratio in croquet strokes, we return to it in section 10.3. For a drive (or stop shot) where Ω_{o} ≈ 0, one may estimate the sliding distance over the grass since by (D1) the speed at which pure rolling begins is (5/7) of the initial speed. Using equation (3) with a typical value of the coefficient of sliding friction, µ_{s} = 0.5, (Hall [4]), the distances needed to reduce the ball speed to (5/7) of its initial value, i.e. 12v^{2}/49gµ_{s}, are 0.05, 0.9, 2.2 and 5.0 m, for A1D  A4D respectively. Except for A1D, these are much larger than the distances measured on the videos, and it is clear that no speed changes (linear or angular) will be detectable on them, but for A1D with total travel roughly one metre, one might expect changes to be easily seen if the ball remained in contact with the ground. In fact no nonlinearity in time could be detected, which strongly suggests that even for this stroke the ball had left the ground over the period of measurement, so that, whatever the origin of its anomalous behaviour, it had occurred almost immediately on impact. 6. Singleball StopshotsThe scatter shots discussed in section 4 above were all made with a stopshot mallet action; attention there was on the resulting ball on ball collision, but the three shots with the largest ball spacings were also suitable for study of the mallet on ball collision, and the results are given in Table 3.
The aim of these strokes was to avoid putting spin on the ball by using negative values of both mallet angle α and stroke angle β, as discussed previously: the measured small values of Ω in row 10 of the Table show that the objective was achieved, and that they compare reasonably well with those calculated from equation (B5), 5, 2 and 6% respectively. The stopshot is certainly not a free swing of the mallet. The video images indicate that the striker can affect its motion within small times after impact, particularly for gentler shots, and one can expect that the values of the mallet speed after impact, V, will be systematically lowered: however, this should not affect the ball speed, and the values of the ratio v_{1}/U in row 8 of Table 3 should be unaffected by what happens subsequent to impact. Equations (A3, 4) of Appendix A show that the predicted value of this ratio is given by
dependent on k and e, but not on the strength of stroke. The measured values in row 8 of the Table are consistent with this with a mean value of 1.25 ± 0.02, and using in equation (4) the value of e = 0.72 from above gives k = 2.7 ± 0.2, in agreement with the value of 2.5 from section 5.1. The ratio of the mallet speeds before and after impact (row 9 of the Table) is V/U = 0.42 ± 0.02, rather than the value of 0.52 expected from the equations of Appendix B for a free mallet swing, suggesting that that the mechanics of the collision has indeed been modified by the stopshot action. 7. Singleball RollsThe single ball rolls (HSAVI, files A1R  A4R) have additional complications over the strokes already discussed because the stroke angle, β, is firmly downwards, leading to strong reaction from the ground both during and after impact, and also to motion in the ydirection, including jumps – which are obvious and impressive in the STDMPEG video. The basic results for the rolls are given in Table 4(a), with all the speeds given as horizontal components of velocity a few milliseconds (ms) after impact: for A1R this is significant since the video reveals a gentle but prolonged further doubletap between about 15 and 25 ms after the first contact. The amount of spin in these rolls is large, Ω ≈ 100%, and all the tabulated parameters for the rolls are greatly affected by the ground. Comparing the values summarised in TEXT with those in rows 4 and 7 of Table 4(a) shows considerable differences; all of the mallet speeds in TEXT are about 20% lower than those measured here, while the contact time of 3.4 ms reported in TEXT for A1R, instead of 1.1 ms here, is a mistake.
Although important, almost nothing is known about the properties of the ground. The collision is certainly very inelastic, and equations (A3, 4) do not apply, even for the horizontal components of motion, as can be seen by comparing the values of the ratios in rows 11  13 of Table 4(a) with the corresponding values for drives in Table 2. In fact, the ydisplacement becomes of crucial importance: the videos clearly show that for v and v* the displacements measured shortly after impact are dependent on the initial drive of the ball into the ground rather than the mechanics of the mallet/ball collision. Fortunately, the video camera was aligned on the upper half of the ball during these strokes, so that although a lawn is not a very satisfactory reference level, the ydisplacement of the ball was fairly easy to find. Table 4(b) summarises the chief results from these ymeasurements, but a much clearer idea of what is happening can be seen from a graph. Figures 3(a) and 3(b) show how the ydisplacement of the ball and its angle of rotation vary with time; figure 3(a) for A4R the strongest stroke, and figure 3(b) for the weakest, A1R, where the resulting jump of the ball can be seen returning to ground level – though the picture here is complicated by the indubitable double tap.
7.1 Spin and Jump in Roll ShotsAppendix C contains a model of roll shots based on the forces illustrated in figure 2(a); however, although useful, the degree of reality which can be attached to it is small because, unlike the drives and stopshots where the ball leaves the ground after impact so that the ground friction terms vanish, for rolls the effect of the ground friction is crucial and not confined to the duration of contact with the mallet. An approximate treatment is given in Appendix C, similar to one used for discussing spin in billiards, which leads to a condition for the production of a state of pure roll, Ω = 1,
This has the solution that the angles α and β should be such that their difference should always be about 22º, and it seems significant that as the strength of the rolls increases the angles in Table 4(a) do indeed move closer to this value. However, the model is really a model of instantaneous impact of idealised, perfect surfaces, whereas figure 3(a) shows that the ball in a roll stroke is so strongly forced into the deformable ground that it rolls along its transient pit and is then launched into a trajectory through the air with a spin which cannot differ appreciably from the condition of pure roll imposed upon it by the extreme frictional force of the ground reaction. The jump is inextricably connected with the high degree of roll, provided that, as here and unlike the twoball rolls discussed later, the mallet does not constrain the striker’s ball by a second impact as it rises from the ground. The duration of contact between ball and mallet is about 1 ms, after which time the ball has acquired a few degrees of spin, and has been driven downwards by about 2 mm; the further depression of 5 mm is caused by the kinetic energy of the ball as it forms a small hollow in the ground – which subsequently recovers, releases some of the energy stored in it, and drives the ball upwards as an aerial projectile moving under gravity. The motion cannot be followed far, but assuming that the almost linear region after about 5 ms is due to flight, the height and length of the jump can be calculated as about 0.3 and 3.6 metres respectively (Table 4(b)); these are impressive distances, and agree as well as can be judged with the overall views of the stroke in the STDMPEG video. The ball has clearly not acquired its full spin during its 1 ms contact time, but rises to its full rate over the following 2 ms as a consequence of its enforced strong contact with the ground. There is extremely strong retarding friction over this period reducing the speed of the ball in order to increase its rate of rotation, cf. Appendix D. Row 9 of Table 4(a) gives the values of the horizontal ball speed in the steady state after the collision and the interaction with the ground, but there was a previous short period where the speed was discernibly falling, and while the accuracy is not high, the measurements indicated the ball speed immediately after impact to be about 10 m/s (i.e. about 10 mm movement in 8 video frames), compared with the final speed of 7.2 m/s, – in good, if fortuitous, agreement with the reduction to 5/7 of the initial speed (i.e. 0.71) for the generation of pure roll from a ball originally sliding without spin. The conclusion is that for singleball roll strokes the amount of roll has little to do with the details of the stroke, provided only that there is sufficient downwards pressure to force the ball into the ground for long enough for friction to impose a state of pure roll onto it. For A4R the value calculated for the fraction of pure rolling, Ω, is essentially unity. The tabulated values for A2R and A3R are somewhat higher, but the factor controlling the spin imposed on the ball during its curved path along its depression is not the horizontal component of velocity, but something nearer its true vector sum, v, as tabulated in row 6 of Table 4(b); if these values are used, the corrected rolling fractions, Ω say, becomes 1.04, 1.05, 0.95, for A2R  A4R respectively, and not significantly different from unity. The transient hollow formed in the ground is surprisingly large, about 50 mm in diameter for A4R. Like any material with some elasticity, the ground stores energy and releases it to send the ball upwards again – there must be some small delay due to energy loss, but this is not apparent in the graphs: the downward velocity of the ball can be estimated to be close to 5 m/s immediately after impact, so comparing this with the upward velocity of 2.48 m/s gives a CoR for the ground of about 0.5. However, lawns can be expected to vary greatly depending on their composition, their grass covering, and their dampness, and, for instance, drop tests in Bristol at a speed of about 6 m/s on a typically grassed, ‘hard’ lawn resulted in bounce heights of about 50 mm, corresponding to a CoR of only about 0.15. The case of A1R is rather different, and shows other features of interest. Much of the jump was within the range of the video, as illustrated in figure 3(b); the results are generally similar to those for A4R, except that in this case the rotation, having risen by about 10 ms to an approximately constant rate, at about 20 ms then falls to a lower rate despite the ball being airborne and moving under gravity in the middle of a jump. In fact, the video shows that the change is the result of a double tap between about 15 and 25 ms after the first impact, as indicated by the shaded area A in figure 3(b). The friction from contact with the mallet face slowed the rate of rotation from a value of Ω ≈ 120% down to the tabulated value Ω = 73%, but apparently only nudged the ball in the xdirection, without affecting its overall average speed. Point B on the displacement graph marks the peak of the jump, and the solid square points calculated for free fall under gravity lie exactly on the measured data, as of course one should expect. The double tap causes the jump to be asymmetrical, and it has also reduced the duration, height and distance of the jump compared with the values given in Table 4(b), which are calculated from the upward speed of the ball before the double tap. 8. Twoball Croquet StrokesAs the videos show, all straight twoball croquet strokes are double taps, sometimes developing into long extended contacts or multiple taps as the mallet follows closely behind the striker’s ball. The first contact time is always close to a millisecond, with the second impact coming very closely afterwards, depending on the strength of the stroke; this is especially true for the roll strokes shown on the DVD, where the control of roll appears to be by fine adjustment of the follow through, no doubt purely instinctive, but hard to reconcile with the requirement that the mallet be not accelerated through the stroke. The shots at different strengths do not form a clearcut pattern, which limits systematic comparison between the different types of stroke: for instance, the distance ratios for the three stopshots vary from 4.1 to 6.2, practically overlapping those for the drives, which vary from 3.0 to 4.0; and while the four full rolls are more consistent, with distance ratios between 1.0 to 1.2, they overlap the socalled passrolls with ratios of 0.9 to 1.2 – though these seem to be only different from the ordinary full rolls in that they have a slight angle of split. Many players can play a ‘super passroll’ with a distance ratio of about 0.5; these are sometimes viewed as dubious, but often enough they are played with a crisp stroke which it would be of great interest to understand in detail. Unfortunately, no such stroke is shown on the DVD. The major difference between the twoball strokes is again in the initial motion of the striker’s ball; ‘uppish’, for drives and stopshots, downwards for rolls, and this is the distinction used below, after a review of the only theory so far which attempts a systematic explanation of the different croquet strokes. 8.1 Calladine and Heyman’s Heuristic ModelBefore further detailed analysis, it is instructive to give an outline of Calladine and Heyman’s (CH in what follows) model for croquet strokes, which is based on the idea of successive, independent, 2body collisions, as in Appendix A. The model is simplest for elastic collision (which suffices to illustrate the basic idea) when equations (A3, 4) become
of which the second is especially instructive as it suggests that the mass ratio, k, may be regarded in a heuristic sense as a measure of control of the mallet during a stroke. CH consider two limits of behaviour: their “stopshot” corresponds to k = 1, when the equations imply that the mallet stops dead while the striker’s ball moves with speed v_{1} = U; the other limit they call a “push”, corresponding to k equal to infinity, when the mallet continues to move with its original speed U, while the ball moves with speed v_{1} = 2U. Considering now the further impacts during a croquet stroke in these two limits: for the “stopshot” the striker’s ball, ball 1, transfers all its momentum to ball 2 which thus moves with v_{2} = U, while ball 1 and the mallet have both stopped dead; for the “push”, ball 2 moves with v_{2} = v_{1} (= 2U), but the instantaneouslyatrest ball 1 is immediately given a second impact by the mallet (still moving at its original speed) identical with the first, and thus follows behind ball 2 with the same speed as it was given in the first impact, v_{1} = 2U. The model thus replicates the limiting behaviour (and distance ratios) for the idealised stopshot and fullroll, while the standard drive corresponds to a value of k somewhere in between. The greatest value of CH’s model lies in its application to split croquet strokes, especially in their calculations for inelastic collisions with e = 0.75, but it says nothing about the roll of the ball which is well known to be an important factor in the control of distance ratio in croquet strokes. CH do discuss spin in their brief account of the jumpshot, and their equation (5) has a similarity with the present equation (C1) (though a printing error leads to a ‘2’ being misplaced), however, they make no use of the equation in their subsequent analysis. No doubt a small measure of control of the croquet stroke is possible by control of the mallet speed during the millisecond or so of the impact (cf. section 6 above), and this can be thought of as equivalent to control of the effective mass ratio k, but for a moderately strong stroke the CH limits would require speed changes of about 3m/s in a time of 1 ms, i.e. accelerations of about 300g, vastly greater than possible for a human player. 9. Twoball Stopshots and Drives9.1. Spin
All the 2ball drives and stop shots (HSAVI files, series C), and close, doubletap scatter shots (HSAVI files, series B) were made with an uppish mallet stroke, i.e. with negative values of the angles α and β, and the results for them are shown in Table 5. The basic data are tabulated in rows 1 to 11, including the rolling fractions for the striker’s ball, Ω_{1}, and also for the croqueted (or scattered) ball, Ω_{2}, where these could be measured; these are the values that might be expected to show significant differences for different types of stroke, particularly for Ω_{1}. The stop and scatter shots tend to have smaller values of Ω_{1} than the drives, but the effect is small compared with the variations for the nominally similar strokes: the effectiveness of a wellmade stopshot is suggested by the gentle shot C3S, with a distance ratio of 6.6 (row 17), but the distance ratios for the other stoptype shots vary between 2.8 and 4.2 in no systematic way. An apparent exception is the result for stroke B1MS (underlined), however, the value of 2.45 m for its travel distance L_{1} (row 8 of Table 5) taken from TEXT is anomalous, and inspection of the STDMPEG video shows that it cannot be correct. Further indication of error comes from the value of lawn friction, µ_{1}, which is about twice what is expected, – a likely explanation is an inversion error in recording, e.g. writing 2.45 instead of 5.42 (a not uncommon accident) which would remove both discrepancies. Overall, one must conclude that the the data do not reveal any systematic differences of spin between these different croquet strokes and thus, unfortunately, do not allow any general conclusions to be drawn from them which could lead to improvements to the CH theory. 9.2. Contact TimesThe contact times for all the strokes in Table 5 are illustrated schematically in figure 4 in terms of the number of video frames of apparent contact. Though often overlapping for the stronger strokes and when the balls are close together, three distinct impacts of the balls can be seen on the videos, each of duration less than about a millisecond; this implies that where spin on the striker’s ball is not important, and where there is negligible vertical movement, the assumption of successive, independent 2body collisions is amply justified, and the strokes can be analysed as were the single ball drives in section 5.1. The 2ball rolls, however, will be considered later. 9.3. Speed RatiosThe simplest prediction of Calladine and Heyman’s model is for the ratio of speeds of ball 2, the croqueted or scattered ball, to the initial mallet speed (i.e. v_{2}/U) after successive applications of equation (A3), with the value of the mass ratio k being equal to unity for the second (ball on ball) collision, while the CoR, e, is the same for both collisions since one expects the energy loss to arise only from the ball. The discussion in Appendix A leads to,
giving a ratio for v_{2}/U which is independent of the strength of stroke. The values of this ratio in row 13 of Table 5 show no clear systematic variation and, within the precision of measurement of about 3% for the speeds, are approximately constant at 1.10 ± 0.02 (sem); thus, using e = 0.72 as found in section 4.1, one obtains a calculated mallet/ball mass ratio of k = 2.9 ± 0.5, higher than that deduced previously, and although only of limited accuracy, perhaps nearer to expectation. When the mallet makes its second impact on ball 1, both are in motion and the notation becomes more complicated; it is convenient to define the speeds of mallet and ball 1 after the second impact as W and w_{1} respectively, when
and
The most interesting results for comparison with experiment are the ratios w_{1}/W and v_{2}/w_{1}, which are evaluated for a range of values of k and e and shown graphically in figures 5(a) and 5(b) together with the experimental values from rows 14 and 15 of Table 5. The ratio w_{1}/W, i.e. final striker’s ball to mallet speed ratio, proves to depend only weakly on k as shown in figure 5(a): the experimental points are plotted on the graph at e = 0.72, as obtained previously, and scatter at about the expected level of random error in the measurements. At this level of uncertainty no systematic differences between the different types of stroke are evident, nor does there appear to be any correlation with the values of Ω for either ball; however, the mean from all ten measured values of the ratio is 1.71 ± 0.05, which compares well with the value of 1.705 ± 0.005 obtained by using e = 0.72 in equations (A9, 10). The model is at least self consistent. The ratio of ball speeds v_{2}/w_{1} (n.b. these do not need correction for roll) from row 12 of Table 5 are similarly plotted in figure 5(b), again at e = 0.72, but in this case the theoretical curves depend strongly on k. Again, no clear distinction can be made between the different types of stroke, and the only useful statistic is the mean value of the ratio, 2.07 ± 0.06: the square of this implies that for a croquet stroke where both balls are without initial roll the distance ratio L_{2}/L_{1} should equal 4.3, which is close to common experience for the standard drive. The speed ratio yields (coincidentally) an effective value of k = 2.07 ± 0.08, surprisingly much lower than the previous values between 2.5 and 3. The square of the ratio of ball speeds corrected for roll, (v_{2}/w_{1})_{r}^{2} (cf. section 5.4) is given in row 16 of Table 5, and is expected to equal the distance ratio of the balls in row 17, L_{2}/L_{1}: on the whole the agreement is fairly good. 10. Twoball Roll StrokesThe basic results for nine rolls from HSAVI series C are shown in Table 6, rows 1 to 9; the TEXT file also lists a gentle halfroll C1H, but this was not found on the DVD. There is special interest in the rolling fractions of the balls, but while Ω_{1} (row 5) can be found easily, data for Ω_{2} (row 8) are limited because the videos do not show sufficient of the motion: in all cases Ω_{2} began by being negative (i.e. backwards roll) after impact by the striker’s ball (cf. figure 1(a)), but become positive as a result of ground friction, though only for C1F was it possible to study this in detail (cf. figure 6 and section 10.2 below).
The most striking feature of the rolls is the prolonged contact, cf. figure 1(a): in TEXT they are all reported as greater than 30 ms, i.e. extending over more than 250 video frames; however, this is slightly misleading. In every case the initial impact can be seen from the videos to last only about 1 ms, as for the singleball rolls, but it is followed by a period where it is impossible to see whether there is contact or not, and values for the mallet speed after impact are not given in Table 6 because they are essentially the same as for ball 1. It is easy to see how the multicontact and streaked patterns that have been reported from carbonpaper impressions of roll shots may arise [2, 7, 8], however, there is no clear evidence of acceleration throughout the apparent contact, so that one cannot conclude that the ball and mallet are indeed touching. All of the twoball rolls appear to have a very similar behaviour, but the video of C1F, the slowest of them, shows the most complete coverage of what is happening, and is illustrated in figure 6. The mallet stroke is similar to that illustrated for the singleball roll in figure 3(b), and the behaviour of the striker’s ball is rather similar too; there is a marked initial penetration into the ground in both cases. For A1R there is no possibility that this was due to anything but the momentum of the ball – there is clearly no contact with the mallet during penetration – but for C1F this is certainly the period when the ball is in contact with the mallet, as discussed further below. 10.1. The Striker’s Ball, Ball 1The xdisplacements of the balls are shown by the downward sloping lines at the top of figure 6. There were significant speed changes for both during the initial few milliseconds, which are most easily seen from the curve which shows the distance between them, labelled ∆x, and shown by solid black squares on the enlarged scale on the right hand axis. For ball 1 there is a considerable acceleration for about 5 ms after the end of the initial contact period, i.e. during the time when the measurements of the ydisplacement show it is moving downwards into the ground; its initial horizontal speed is reduced by impact with ball 2, but is almost immediately accelerated by the second contact with the mallet which is clearly visible on the videos  the mallet and ball must be in contact during this acceleration. However, the subsequent upward motion of the ball (which leads to prominent jumps for the singleball rolls) is restrained by the downwardsloping mallet face, which results in the ball rolling/sliding up it. All this is happening within a few ms of impact, after which there appears to be no further change of ball speed. The rotation measurements ø(t) also shown in figure 6, show that during the same few milliseconds when the ball is pressed into the ground, the angular rotation rises to a constant rate which proves to correspond to Ω = 1, i.e. to perfect roll. The precision is not high, but there is no sign of any further impulse from the mallet over the rest of the period of apparent contact, either in the xdisplacement or in the rotation (cf. by contrast the results for stroke A1R shown in figure 3(b)), so the period of extended contact for this stroke appears to be ≈ 5 ms, rather than the 30 ms given in TEXT. The pure roll imposed on ball 1 within about 5 ms of the initial impact implies that it should thereafter suffer a constant retardation due to the rolling friction µ_{r}mg, resulting in an xdisplacement which is quadratic with time; however, even if it were certain that the ball was in contact with the ground, something impossible to tell from the videos, the upward curvature that this would produce is so slight that it could not be seen over the ≈ 50 mm of travel on the video. 10.2. The Croqueted Ball, Ball 2During what is always a short collision time between them, ball 1 necessarily causes ball 2 to rotate with initial backward spin, and also, though it is not possible to follow it on the HSAVI videos, gives it a downward motion, cf. Appendix B. One therefore expects ball 2 to show a reduced version of the behaviour of ball 1, and suggestions of the resulting minor jumping can be seen in STDMPEG – and are often seen for roll strokes in ordinary play. The downward motion causes ground friction to act on ball 2 after it has moved away from ball 1, and gives rise to the deceleration which is shown up to about 15 ms in figure 6. The angular speed of ball 2 starts off negative, with a value of about 300 °/s, but over the same time as its linear speed decreases due to ground friction, its angular speed increases to about +500 °/s. The changes of the linear and angular speeds of a ball under a common surface force are necessarily related by equation (B6) of Appendix B, and writing this in terms of the measurements x(t) and ø(t), one finds that their nonlinearities in figure 6 should be eliminated by plotting the composite quantity (x + 0.321ø) against time: although the precision is not high, the prediction is confirmed within the uncertainty of measurement, which confirms that both of the nonlinear changes up to about 15 ms are the result of the same ground frictional retardation. The angular speed of ball 2 appears to remain constant after about 15 ms, at a rate corresponding to Ω_{2} ≈ 0.4: if the ball had remained in contact with the ground, it would continue to experience an accelerating force until Ω_{2} = 1, but since it did not, there is clear evidence that it had left the ground at this point in response to previous compression into it. The curvature in x(t) and ø(t) up to ≈ 15 ms allows an estimate to be made of the coefficient of sliding friction, since both are caused by the frictional force µ_{s}R2, where the impulsive reaction R2 arises from the downward component of momentum as the ball first presses into and then rises from the ground. The curvature is more obvious for ø(t), but the calculations are simpler for x(t) and these are given here: the data are very approximate, but figure 6 shows that during the time when the force is acting the value of the horizontal speed, i.e. v_{2} cos(δ), has decreased by about 0.5 m/s, while R2 changes by ≈ 1.5v_{2} sin(δ), where δ is much the same as the initial stroke angle β, and where the multiplier 1.5 (rather than 2) takes account of the loss of energy as ball 2 bounces upwards, cf. figure 2(b). On substituting for v_{2}, we find that µ_{s}tan(δ) ≈ 0.3, and since δ ≈ 16°, we finally deduce that µ_{s} ≈ 1.0: this is double the value of 0.5 obtained under very different conditions by Hall [4], and about fifty times the value of µ_{r} calculated from the rolling data for C1F (cf. Table 6). To put these coefficients into perspective, one can calculate that if, instead of jumping, the ball had rolled on the ground held down by its weight, i.e. R2 = mg, then it would have taken another 15 ms to have achieved ‘pure roll’ under the action of sliding friction, and another 3 seconds to come to rest under the action of the much weaker rolling friction, rolling distances of about 12 and 1000 mm respectively. 10.3. The Distance Ratio and Lawn Friction, µ_{r}The distance ratios L_{2}/L_{1} for the full and passrolls in Table 6 all have a value of about unity; both balls have travelled nearly the same distance, and any irregularities in lawn behaviour will tend to cancel out, cf. equation (3) of section 5.4 – though using v_{2} and w_{1} in the present notation. The amount of roll is critical for the stroke, and it is essential to make proper allowance for spin by using equation (D1) when calculating the distance ratio. Inspection of the relevant rows in Table 6 shows that the correction makes a large difference, and results in the calculated and measured values (rows 11 and 12) being in rather good agreement, with the differences at the level of the measurement uncertainty. The agreement is not so good for the two halfrolls, or for the strokes in Table 5; some of the reason for this is probably that the balls in those cases were not rolling over the same pieces of grass and not experiencing the same retarding force, and this is manifested as variation of µ_{r}, the coefficient of lawn friction, as we now discuss further. Lawn speeds are certainly known to vary, but, unsurprisingly, have not been studied in detail. Figure 7 contains all the values of µ_{r} calculated using the corrected form of equation (3) plotted against the ball speed. The scatter of data is considerable, about three times more than expected from the random errors of measurement: changing lawn conditions throughout the experiments, and also the traffic over it, no doubt contribute to the extra scatter, but whether they account for all of it, and whether the results are typical of an ordinary club lawn remain to be discovered. Of the two big outliers in the top left corner, the open circle is the known error in B1MS (cf. section 9.1 above), and while no error is known for the solid circle for A2R, one may conjecture a mishearing of spoken measurements e.g. “7.15” (as in TEXT) misheard for “11.15”, which would remove the anomaly. However, despite the scatter, the figure shows an unmistakable increase of µ_{r} for higher speeds which does not seem to be correlated with the type of stroke, i.e. with the roll on the ball. The systematic increase of µ is shown in figure 7 as a variation with ball speed, but it could be that the distance travelled is the more significant variable. Some such effect might be expected if the grass on a lawn is less trodden for the longer strokes, but this is conjectural. It is also the case, as pointed out in the discussion of equation (B6) in Appendix B, that even in pure rolling, the gradual slowing down due to ground friction necessarily involves an element of sliding friction, and it may be conjectured that this is more important for the stronger strokes. It is certainly the case that strong strokes often have an easily visible small scale bumping motion across even a good quality lawn, and it seems likely that the resulting friction will be some average value between zero when the ball is in the air, and the necessary episodes of very high sliding friction. A dependence of the coefficient of friction on speed is not normally to be expected, but this is partly at least due to the fact that it is not commonly studied; however, recent work by Cross [11] shows that it can easily be observed in simple experiments – though these were not on such illdefined materials as the surface of a lawn. There is much that is unknown about ground friction, but little incentive to find out more: no doubt the effect, if confirmed, is significant as it must certainly affect the distance ratio of croquet strokes, but no doubt ordinary croquet players are quite unaware of it and subconsciously adjust their strokes in the light of experience. 11. Summary and ConclusionsThese HSAVI videos are important as the first of wide scope at speeds high enough to resolve details of impact – but like much pioneering work, they leave room for improvement. The images show only a small part of the ball(s), are often without clear reference marks, and often have shadows, poor visual contrast, and parallax and focus problems. No doubt these things were due to limitations of the technology and of the venue, but for serious further work they need to be addressed. The methodology too was flawed by the (understandable) choice of a human to play the strokes since in the event no clear distinction can be made between many of them, nor was it possible to repeat strokes ‘exactly’, in order to study their natural variability. Controlled experiments they were not, and no important conclusions emerge from this analysis – which is perhaps to be expected as they were not made with this end in view. Nevertheless, as they have opened a new realm of measurement, it seemed important to study them in depth to see what they contain.
In conclusion, there are several things in the videos to intrigue a physicist, but it is hard not to share Newton’s feelings when he wrote, [12], “… but the examining severall thinges has taken a greater part of my time then I Appendix A. Simple Collision(i). If in figure 2(a) there is no motion in the ydirection, if the angles α and β are small enough for their cosines to be approximated by unity, and if the effect of ground friction can be neglected (as for instance for the drives in section 5), the collision can be analysed using the equations for particle dynamics in one dimension (see e.g. [9], section 10.3). It is convenient to consider first the case where both mallet and ball are initially in motion, with speeds of U and u respectively, when their speeds after collision are given by V and v, where
and
k is here the mass ratio of mallet to ball, and e is the coefficient of restitution (CoR). When the target ball is at rest, u = 0, and equations (A1) and (A2) simplify to
and
The coefficient e is a measure of energy loss due to inelastic deformation on impact, nearly all in the ball, and the relative loss of kinetic energy, ΔE/E, due to impact can be calculated from (A3, 4) to be given by
for the collision of two croquet balls this is close to 25%. If any of the initial kinetic energy is diverted away from appearing as kinetic energy of rectilinear motion (e.g. as by roll, cf. section 5.3), the relative shortfall of kinetic energy, δE/E, gives rise to an apparent change in the value of e, δe, which can be found by differentiating (A5), i.e.,
(ii). The treatment of croquet strokes by Calladine and Heyman assumes that the 3body interactions of croquet strokes can be modelled by successive 2body collisions, without spin, as above. The first impact of mallet on the striker’s ball (ball 1) is thus described by (A3, 4), while the collision of ball 1 with the croqueted ball (ball 2) immediately afterwards is given by
since ball 1 is providing the driving impetus and k = 1 for this collision. This gives
When the mallet makes its second impact on ball 1, both are in motion and (A1, 2) must be used. The notation becomes more complicated, and it is convenient to define the speeds of mallet and ball after the second impact as W and w_{1} respectively. Straightforward algebra leads to
and
expressions which are not transparent, but which can be evaluated numerically and presented graphically as in the main text. Appendix B. Spin in ‘Uppish’ Strokes.(i). The terms ‘spin’ and ‘roll’ have been so far been used almost interchangeably; both refer to rotation of the ball, but while ‘spin’ is a property of the ball, ‘roll’ generally refers to the process by which spin is applied, usually by the effect of ground friction. Figure 2(a) shows an impulsive force F(t) acting during a downward stroke on a single ball at point C determined by the mallet angle, α, while its direction is the stroke angle, β, both measured with respect to the horizontal. It is convenient to resolve the forces at A and C into the components normal to and parallel to the tangents at the points of contact; the normal forces are thus R1 at the ground, and R1* at the mallet face. If there is relative motion between the surfaces in contact at either A or C, each may give rise to frictional forces, µR1 and λR1* respectively, whose direction always opposes the relative motion. The weight of the ball is shown on the figure, but its size of 4.5N is negligible compared with the average size of the impulse during even the weakest drive (≈ 500N) and it is omitted in the treatment below – though important of course in finally bringing a ball to rest. Both sources of friction arise in general, and exact analysis is impossible because the contacts are not points but illcharacterised deformable media, and also because the values of µ and λ are not known. However, there are some cases where only one source of friction is acting, and for these fairly realistic physical models can be devised: the spin from mallet friction is considered here, and the roll from ground friction in Appendix C. The normal reaction at A due to the impulsive force F(t) at C is R1 = F(t)cos(α  β)sin(α), which is positive or negative depending on the sign of α; however the reaction at A also depends on the extent to which the parallel component of force at C, i.e. F(t)sin(α  β).cos(α) is transmitted to the ground via the contact at C. If, at one extreme, the whole of the parallel component were to be transmitted, then the net reaction at the ground would be F(t)sin (β): at the other extreme – corresponding to a frictionless pointmass collision as in Appendix A above, no contribution from the parallel component would arise, and the reaction would be F(t)cos(α  β)sin(α) as before. A real mallet stroke is considerably more complicated than either of these limits, and in general, the reaction depends on both α and β, though usually more strongly on α: strokes where α and β are such that R1 is zero or negative, so that no ground friction arises, and are here called ‘uppish’ strokes (cf. also below). For an ‘uppish’ stroke, positive or ‘forward’ spin is given to the ball when the mallet face has an upward component of velocity at contact which causes the frictional force λR1* opposing their relative motion to exert a torque about the centre of the ball. The situation is equivalent (though opposite) to how loft and backspin are given to a golf ball, as explained by Daish [9, section 14.10]: his treatment is directly relevant to croquet, and is outlined below in a croquet context. For the ball at rest, the initial relative velocity between mallet and ball parallel to the face at point C is Usin(α  β), and the resulting force from sliding friction, λR1*, accelerates the centre of mass of the ball parallel to the face at a rate given by λR1*/m until their relative velocity is reduced to zero and the frictional force disappears: the velocity perpendicular to the face is of course unaffected. When sliding ceases, the spin of the ball is such that its surface is moving with a speed, v*, given by the frictional impulse integrated over the duration of the sliding time, T_{s}
The same force also produces a torque of rλR1* about O, giving an angular acceleration of rλR1*/I, where I is the moment of inertia of the ball about its centre, 2mr^{2}/5, and at the end of sliding the angular velocity, ω (in radian measure), is given by an expression involving exactly the same integral,
Eliminating the integral between (B1, 2), substituting for I, and recalling that since when rolling begins v* = rω, (B1, 2) can be rearranged to give,
independent of λ, the coefficient of friction between the surfaces. Now, from equations (B1, 3), the value of the impulse integral up to the time when sliding ceases must be 2mUsin(α  β)/7λ, while the impulse in the direction perpendicular to the mallet face and responsible for the momentum imparted to the ball, mv, is exerted over the whole time of contact, T: for the rolling motion to be fully established we require T > T_{s}, i.e. that
For the drives studied here, the rhs of (B4) is ≈ 0.05, and although λ is unknown for the mallet face, typical values of sliding friction are ≈ 0.5, and the inequality is surely satisfied. The maximum distance the ball could slide during contact (i.e. if λ = 0) is TUsin(α  β), between 0.2 and 5 mm downwards for the drives studied here, though for realistic values of λ it should slide much less. One can conclude that the spin of the ball is fully established within a small fraction of the contact time, and with rather small movement across the mallet face, so that, using (B3), the rolling fraction Ω can be written as
The model is a simplification since the video images show that the contact is far from pointlike, and give no evidence for sliding along the mallet face. In fact, Hertz’s theory, cf. [6] shows that the compression noted in section 3.1 for A4D produces a flat contact area of of about 12 mm radius, as typically seen by carbon paper impressions: the energy stored in this compression zone is responsible for the linear motion of the ball after impact, and the spin presumably arises from energy of shear which is also stored there. But there is also an important difference between the golf and croquet cases. The upward sliding along the club face associated with backspin of a lofted golf ball is not limited; by contrast, the downward sliding associated with forward spin in a croquet drive may be restricted by contact with the ground: ground friction then necessarily becomes involved, cf. Appendix C, and this may well be the explanation for the anomalous behaviour for stroke A1D discussed in section 5.3. An important consequence for the state of pure roll follows from taking the time derivatives of (B1, 2) and eliminating λR1*: on substituting for I and remembering that v* = v here, one finds that
i.e. the changes of linear and angular velocity are oppositely correlated. However, there is a paradox in this equation: in the steady rolling state both v and ω do finally decrease to zero, which implies that there must always be some component of slip at the contact with the ground; cf. also Appendix D. (ii). The forces acting on the second ball in a croquet stroke are shown in figure 2(b). Ball 1 is now analogous to the mallet and is moving with velocity v_{1} at angle α*, which is probably much the same as the initial mallet angle α. A complication is that ball 1 has acquired some spin, so that its surface is moving with speed v_{1}* where it touches the croqueted ball; however, the angle which would correspond to α in the previous discussion, is zero here. For ‘uppish’ strokes where the ground reaction R2 disappears, the behaviour of ball 2 is similar to that predicted by equation (B5) above, but with U = v_{1}, α = 0 and β = α*, but also with some probably rather weak dependence on the value of Ω1. For rolls the strong downward impulse gives rise to motion at an angle δ similar to though not the same as the angle α*, and thus necessarily an appreciable upward reaction at R2; however, for the angles used in common croquet strokes (hammer shots are not considered here), the ratio of the components of the impulsive interaction at the contact C2, R2*, depend on tan(δ), and are such that the downward frictional force on ball 2 as ball 1 slides over it, λR2*, considerably exceeds the lawn friction, µR2 (equal to µMg, initially at least). Under these conditions ball 2 begins to rotate with backward spin given approximately by equation (B5) again, though after what is always a short collision time between the balls (< 1 ms) this is immediately followed by angular acceleration towards positive values as the ground friction becomes dominant (cf. Appendix C below) – at least until the reactive impulse R2 causes the ball to leave the ground, cf. the discussion in section 9.1. Appendix C. Spin in Roll StrokesIn contrast to the ‘uppish’ strokes where the spin of the ball is produced by the mallet impact, the results presented in the main text show that the spin produced by roll strokes is not a result of the initial impact, but by the strong frictional force from the ground which occurs afterwards: even if a convincing theoretical model could be devised for the spin produced by the initial impact, there are no data available to compare with it. However, it is instructive to consider a simple, idealised model which constrains the mallet friction to produce no torque, so that roll is imparted to a ball by ground friction only. Consider again the impact illustrated in figure 2(a), and assume that the impact at C is such that there is no relative motion between mallet and ball so that, although friction exists at the contact, it does not produce torque on the ball. The only source of spin is now the ground friction at A. The situation is similar to that in Appendix B, and an analogous impulse calculation gives
(which can be extended to include mallet friction, when similar trigonometric terms involving λ also enter, though not appearing to add anything useful). For the special case when roll begins without sliding (i.e. v and v* are equal and opposite) there is no relative motion at point A so that all the terms involving µ must drop out, giving
and when Ω = 1, we have
This is an interesting result familiar in the context of billiards, cf. [9, section 14.5], and also useful for thinking about croquet. When applied to billiards, the cue angle (and line of impact) β is assumed to be zero so that the condition for pure roll becomes sin(α) = (2/5), or α = 23.6º – a result which is geometrically the same as a horizontal cuestroke at a height of 2/5 of the radius above the centre of the ball. Equation (C3) can be solved for other values of β, but even for β = 30º the difference (α  β) has changed only from 23.6º to 20.3º. For practical croquet, equation (C3) is a useful guide on how to produce roll. Equation (C2) is evidently different from (B5) obtained from a different model of behaviour: in (B5) the ball is free to respond to forces applied by the mallet, but in (C2) there is besides the mallet impulse, the constraining effect of the ground, though as the ground is in fact a deformable medium, the distinction is probably not clearcut. The disappearance of both ground and mallet friction in (C2) is the result of assuming that there is no relative motion at the two regions of contact, but how good the approximation is, is impossible to say. It would require the friction forces to be so high that no relative motion can occur – and this indeed is often hinted at in measurements of friction, where an initial ‘stiction’ phase of much higher frictional force is often observed before dropping down to the normal levels when sliding begins: in billiards this is promoted by ‘chalking’ the tip of the cue. The duration, and even the existence, of a stiction period would presumably depend on the deformation of the surfaces in contact and on the reaction forces between them, but it appears not unlikely that even when the conditions leading to equation (C2) are not strictly met, nevertheless it will contribute something to the spin acquired. The treatment of frictional force has throughout been based on the usual empirical form, as the product of the normal force at the contact multiplied by a coefficient of friction, e.g.as µR1 and λR1* in figure 2(a), and µmg in the derivation of equation (3) of the main text. To what extent this is realistic for ground friction is hard to say, and in fact the results of section 10.3 cast considerable doubt on it; however, until much more is known it is difficult to see what else can be done – and hard to see how a truly complete understanding of the physics of croquet strokes can be obtained. Appendix D. The Onset of Pure RollThis is discussed in [4, 10], but is here extended to include the case where one knows the initial rate of roll of the ball in addition to its initial linear speed. Pure roll occurs when the surface speed of the ball, v*, is equal to its linear speed, v, so that no slip occurs between ball and ground. Defining the relative rate of roll by the ratio Ω = v*/v, the condition for pure roll is Ω = 1, and the coefficient of rolling friction, µ_{r}, is then relatively low. In general, Ω≠ 1 immediately after impact, and if the ball is touching the ground, slip then occurs with about a ten times greater coefficient of sliding friction, µ_{s}, Hall [4], and causes both v and Ω to change rapidly until Ω = 1. During sliding, friction at the ground opposes linear motion, but also produces a proportionate torque on the ball which increases its rotation, as in equation (B6) above, and since v* = rω, this gives the speed at the onset of pure rolling, v_{r} say, as
where v_{o} and Ω_{o} are the initial speed and the corresponding initial relative rate of roll: equation (D1) is true for any value of the frictional force. For an initial pure (forward) roll with Ω_{o} = 1, v_{r} and v_{o} are evidently equal, but when the initial value of Ω_{o} is close to zero, e.g. after a drive, the error in calculating µ_{r} from equation (3) without correction to v_{r} is close to 50%. By contrast, because µ_{s} is much larger than µ_{r}, its effect on the the distance the ball travels before coming to rest is only a few percent, which is unimportant at the present level of accuracy – the distance could be corrected too, if µ_{s} and its possible variations (including zero if the ball becomes airborne) were known, but in practice this is never worthwhile. All the values of lawn friction tabulated in the text have been calculated from equation (3), including the correction for spin using equation (D1). High speed videos: https://www.youtube.com/watch?v=M4EH7Omxkc8 Single ball shots [1]. C.R. Calladine and J. Heyman, The Mechanics of the Game of Croquet, Engineering, 1962, pp 8613; www.oxfordcroquet.com/tech/calladine/index.asp [2]. J. Williams, Video/carbon images of croquet strokes, 2001; www.hep.manchester.ac.uk/u/jenny/kiwi/sport/croquet/video/video.html [3]. R. Kroeger, Various Croquet videos, 2004, 2005; www.croquet.org.uk/tech/exp.html [4]. S. Hall, When a Mallet Strikes a Ball, 1998; www.oxfordcroquet.com/tech/hall/index.asp [5]. I.R. Plummer, Multiple Taps in Croquet Strokes  Electrical Conduction Tests, 2000; www.oxfordcroquet.com/tech/multiple/index.asp [6]. D. Gugan, Inelastic collision and the Hertz theory of impact, Am. J. Phys, 2000, pp. 9204; www.oxfordcroquet.com/tech/gugan/index.asp [7]. I.R. Plummer, Simple Impact Measurements, 1998; www.oxfordcroquet.com/tech/impact/index.asp [8]. J. Riches, Double Hits in Pass Rolls, 2004; www.oxfordcroquet.com/tech/riches/index.asp [9]. C.B. Daish, The Physics of Ball Games, E. U. P., 1972 [10]. D. Gugan, Experiments on the Game of Croquet; Bouncing and Rolling Balls, 2002; www.oxfordcroquet.com/tech/gugan1/index.asp [11]. Rod Cross, Increase in friction force with sliding speed, Am. J. Phys, 2005, pp. 8126; www.physics.usyd.edu.au/~cross/PUBLICATIONS/30.%20FrictionvsSpeed.pdf [12]. I. Newton. cf. R. S. Westfall, Never at Rest, 1980, C. U. P., p 409 Don Gugan 17.iii.09 All rights reserved © 2009
