Abstract: Jaques Eclipse and Barlow GT balls were struck with a Perspex-faced mallet and the contact with the mallet face recorded using carbon paper. The impact marks showed that the contact area is less for Eclipse balls than Barlow GT's and the deformation (flattening) of the ball on impact is about 1mm. Impact marks were recorded for single-ball and croquet strokes. Of the croquet strokes, only roll shots showed  multiple contacts and sliding of the ball over the face of the mallet. There is however no evidence of 'pushing' or 'pulling' the balls as thes stroke were played.

Introduction.

Using only very simple techniques it is possible to deduce some of the details of when a mallet hits a croquet ball. Experiments were done on the Oxford lawn which at the time was quite lush. A flat-faced mallet was used with 1/4" thick Perspex striking faces. This is a hard surface which does not readily deform - it is impossible to make any impression on the sharp edge with a finger nail. The Barlow balls were from the GT series marketed in 1997/8 and the Jaques Eclipse were some of the final batch released before their fire. The latter was a match set and the milling was as new.

Experimental

Standard white laser-printing paper was sandwiched between the mallet face and modern plastic-film carbon paper. This was stretched over the mallet face and taped in place. Hard strokes were done across the diagonal of the court and the paper sandwich moved after each stroke. As an incidental side effect, this technique is an excellent way of determining whether you are striking with the centre of the face of the mallet. Many sheets showed evidence of hitting the balls well off centre.

Single Ball Stokes

	Figure 1  shows the result of three hard single-ball strokes on a Barlow GT which sent the ball from corner 4 to corner 2. As can be seen the images show the milling crisply. The darker areas are where the carbon paper has delaminated resulting in a thick layer of carbon on the paper. The diameter of the marks is approximately 22mm.
	Figure 2. Hard single-ball strokes on Jaques Eclipse balls. In case B the hit was off-centre and lies over the edge of the face. The diameter of the impacts is approximately 17mm. They are smaller than those of the Barlow under similar conditions.
	As a simple assumption it is taken that the ball just flattens to conform to the face of the mallet. In reality the ball may markedly deform from a sphere. Other experiments would be needed to show this.  Figure 3. The deformation, or flattening, can be calculated from the relationship shown in the diagram opposite. For chords a1 & a2 and b1 & b2 the following applies: a1*a2 = b1*b2. The chords may cross at any angle and do not have to lie across the diameter. In this analysis however the chord b1 & b2 is taken to be the diameter of the ball. The results are given below - the calculation is in the appendix.

These numbers do not tell us much about the balls however. A steel ball bearing and a rubber 'power ball' both bounce extremely well from a hard surface, possibly to similar heights but the deformation will be quite dissimilar. In the case of Barlow balls the ball is homogeneous and moderately soft. For the Jaques it has a hard casing and a bouncy core.

Dick le Maitre [report Dec. 1986]  has measured the milling profile of a Jaques ball:-

The milled grooves in Jaques balls are sloped at ~30' to the diametral plane, so that the total subtended angle from groove side to groove side is about 60'. Average groove depth 1.1mm. Width of grooves from ~0.5mm to 1.5mm.

Thus the deformation is in the order of the depth of the milling. Clearly the deformation must involve flexing of the whole casing otherwise the milling pattern would be crushed out.   There is evidence of the collapse of the milling inasmuch as the white tracks between the milling impacts are much more closely spaced in the centre of the impact. A geometric projection of the milling on to a flat surface would show the opposite; the tracks in the milling would be widest in the centre and become narrower at the periphery. (For a more marked example see Figure 4C)

The impact marks show that there is a single impact and there is no evidence for the ball being pushed along the grass causing it to rotate. Were this the case then there would be a blurring of the pattern caused by the ball rotating against the mallet face. This is entirely consistent with the results of Stan Hall on ball-mallet impact times, and those conducted by the author. These show that the ball is in contact with the face of the mallet for approximately 1 millisecond, during which the ball travels 3-5mm. The ball motion is basically a skid.

A final trivial point is that it would be unwise to have a mallet face which was less than ~22mm in diameter, firstly because of the chances of not making full contact with the ball and secondly because of the chances of cutting it with the edge of the face.

Croquet Strokes

The interest with croquet strokes is in multiple taps: Law 28. FAULTS (6th ed.) states

a. DEFINITIONS A fault is committed if, during the striking period, the striker:

7. subject to Law 28(d), maintains contact between the mallet and the striker's ball for an appreciable period when the striker's ball is not in contact with any other ball or after the striker's ball has hit another ball;

8. subject to Law 28(d), strikes the striker's ball more than once in the same stroke or allows the striker's ball to retouch the mallet

The carbon paper impact tests should show multiple impressions for multiple taps and smears for pushing and pulling. Croquet strokes were played on the Oxford lawn using the Perspex-faced mallet and new Jaques balls. A collage of the results is shown in the figure below.

Figure 4 Croquet strokes played with new Jaques balls.

Clearly the roll shots require some interpretation. The hard drive, C, (such as in a four-ball break placing a pioneer on hoop 3 and joining pivot), and the stop shot, D, yield clear images without any multiple images or smearing. Figure 4C shows the flattening/compression of the milling remarked on above. The stop shot is played with the lower part of the face of the mallet hence the truncated image.

The roll shot was played by angling the face of the mallet by around 30' towards the ground, standing over the touching balls and  holding the mallet shaft close to the head whilst playing a long sweeping stroke. There are other techniques for playing these strokes which may result in different imprints. The roll shots show a large imprint followed by one or more smaller prints, then a long smear. In making these prints the carbon paper film was torn with the pattern of the milling in it. This is consistent with the ball rotating against the face under pressure.

If this is taken to be the order of events, it would be consistent with a hard initial contact as the mallet has to overcome the static friction and inertia of the pair of balls. This yields the large diameter imprint. The mallet is decelerated by the collision and the balls start to move. Whether the mallet separates from the balls at this time cannot be deduced without further experimentation. The mallet then presses into the moving balls, producing the smaller multiple images (possibly with some chattering). Finally as the mallet decends, due to completing the lower part of the arc of its swing, the balls grate against the face leaving an upward smear. This is however speculation.

What is clear is that there are multiple hits; whether they are audible and distinct (28.a.4. moves the striker's ball other than by striking it with the mallet audibly and distinctly; ) as regards practical refereeing is another matter.

The rush shot only shows a single impact. It was speculated that there might be multiple contacts with the striker's ball oscillating between the roquetted ball and the mallet face. A multiple hit is not a fault in a direct roquet.

Conclusion

For single-ball strokes the impact marks showed that the contact area is less for Eclipse balls than Barlow GT's and the deformation (flattening) of the ball on impact is about 1mm. The imprints show general flattening of the balls but also of the milling. The flattening is not such as to totally crush the milling texture. All single-ball strokes produced clean images not showing any evidence of the ball rotating whilst in contact with the face.

Of the impact marks recorded for croquet strokes, only on roll shots were multiple contacts and smearing recorded. This is indicative of the mallet rebounding from the balls and subsequently the striker's ball rotating against the mallet face. The flattening of the milling was more marked than for single ball shots due to the increased mass of the pair of balls being struck  instead of a single one.

 Dr Ian Plummer
Balliol College
Oxford, October 1998

Appendix: Calculation of flattening of sphere.

Referring to the notation of Figure 3;
a₁ * a₂  = b₁ * b₂. b₁ & b₂ lie on a diameter, hence b₁+ b₂ = D, the diameter of a croquet ball, and a₁+a₂ = d, the diameter of the impact mark. This is sufficient information to set up a quadratic which can be solved for b₁.

a₁+a₂  = d, a₁ = a₂, hence a₁ = a₂ = d/2
a₁ * a₂ = b₁ * b₂
d²/4 = b₁ * b₂
b₂ = D - b₁
d²/4 = b₁ * (D - b₁) = -b₁² + D*b₁
b₁² - D*b₁ + d²/4 = 0

This last equation is quadratic in b₁ and can be solved using the standard formula; ax² + bx + c = 0

 x = (-b +/- sqrt(b²-4ac))/2a
D is taken to be 92.075mm (3 ⁵/₈").
a₁ = 11mm (for Barlow)
a = 1
b = - D
c = d²/4
b₁ = (D +/- sqrt( D² - 4*d²/4))/2
 b₁ = (D +/- sqrt(D² - d²))/2