Basic Theory of Dynamics

There are a number of questions, such as

What angle will the roqueted ball travel after a cut rush? And how far?
What angles and directions will the balls travel after a croquet shot?
How much deflection can I expect if I hit off-centre?

that can be answered either by experience of playing many times, or by the theory of dynamics. If the theory of dynamics happens to interest you, it might be rewarded, partly as a matter of interest, partly because it can alert you to rules of thumb that usefully supplement and illuminate practical experience, and partly because it answers some questions about how different mallets perform.

Acceleration and Force

A stationary body cannot move unless it is subjected to a force. While subject to a force, it will accelerate in the direction of that force. This can be put either in the form of Newton’s laws of motion, or as the more vernacular observation that, if you whack something, it will tend to move off in the direction you whack it.

There are many ways of applying force to an object like a croquet ball. One way is to hit it with something else, like another croquet ball or a mallet. As it happens, we know how long the impact lasts when a mallet hits a single ball; it is about a millisecond, regardless of how hard you hit it. People have taken videos with high speed cameras to show this. So for about half a millisecond, the ball gets compressed around the area of impact, and then for another half a millisecond decompresses as the material’s resistance to being swashed forces the two bodies apart. In a long shot, the force is quite considerable: enough to accelerate the ball from a standing start to, say, 50 m.p.h in a millisecond.

However the force is applied, away we go. The more force that is applied, and the longer it is applied, the more the acceleration. There are equations that can be used to work out the relationships between speed, force, time and distance.

Furthermore, these equations work separately in each of the 3 dimensions. For most purposes, motion up and down is irrelevant to croquet – the balls tend to stay on the lawn, more or less. That leaves two dimensions. So, if you are hitting up the lawn, there are calculations that can be done to work out the motion along the North-South axis, and separate calculations for the motions along the East-West axis.

The 4 Equations of Motion

There are 4 equations often used to calculate the relationships between force, speed, distance, time and uniform acceleration; each of them have a common sense expression:

v = u + ft	The speed something will end up going is the speed it was going before plus the amount of force applied times the period of time it is applied.
s = ut + ½ ft²	The distance something will go during acceleration is however far it would have gone anyway bearing in mind its initial speed (if any) plus half of the amount of force times the square of the time
v² = u²+2fs alternatively expressed as f = (v²–u²)/2s	The amount of force needed for the acceleration is the difference between the squares of the terminal velocity and the initial velocity, divided by twice the time
s = ½(u+v)t	The distance traveled is the average speed times the time
where v is the velocity at any given time u is the velocity before the force is applied f is the force applied t is the time elapsed, and s is the distance traveled

The Conservation of Momentum

The momentum (we are talking about linear or straight line momentum for the moment; we get to angular momentum presently) of a body is its mass times its speed. The law of conservation of momentum says that the momentum of two or more unrestrained bodies after an impact is the same as it was before. That holds good even if the bodies are not perfectly elastic (well, nothing is). And again, it works separately in each dimension.

Rushes

So take one croquet ball hitting another. Assume blue is struck due north, and hits red full on at, say, 10 feet per second. Now, the law of conservation of energy says that the momentum before impact is 10 lb ft sec^-1 (there is no prior momentum in the red ball, since it is not moving before the impact, so the only prior momentum is that of the 1lb blue ball traveling at 10 ft/sec). Now, if the balls were perfectly elastic, the blue ball would stop dead in its tracks. And so the red ball would move off with a momentum of 10 lb ft sec^-1. Since its weight is the same as the blue ball – 1 lb – it will thus move off at the same speed as the red ball was going. And so the red ball would end up precisely where the blue would have gone.

But croquet balls are not perfectly elastic, and the blue ball will roll forward a little after impact. And here comes the first conclusion. The speeds, and hence the distances traveled by the red ball and the blue ball after impact are the same as the speed of the blue ball before impact, and hence the distance that the distance that the blue ball would have gone but for the impact. Put another way, if the blue ball dribbles on a yard past the point of impact, the red ball will end up a yard short of where the blue ball would have ended up had the red ball not been there. It worth bearing in mind that this rule works whatever the elasticity of the balls. It works for Barlow balls, Dawson balls, whatever, as long as they weight the same. Certainly it is true that the rushes change according to the type of ball. If you are used to Dawson International balls, then Barlow balls will seem “doughy”.

Take Offs, splits and pass rolls – sideways movement

Now, here is an interesting thing about any croquet shot. Imagine the line of the swing of the mallet (it might be north-south, but will not matter if it is anything else). Immediately before impact, the mallet will have lots of momentum north-south, but none east-west. And of course the balls are stationary, so they have no momentum at all. Assume the shot is “clean” so that after the impact, the mallet has slowed down, but does not slew off to either side.

So: before the impact, the east-west momentum is nil. The total east-west momentum after the impact must also be nil. There is none in the mallet head. The mass of the two balls is the same. So, the speed at which one ball is moved to the left is the same as the speed at which the other ball is moved to the right. Thus, roughly (not quite exactly for a variety of small effects), the distance the striker’s ball will move to one side of the aim line is the same as the distance the croqueted ball will move the other side.

So:

In the case of a take-off – or “fat” take-off – the croqueted ball will move out the same distance as the distance between the aim point and the resting point of the striker’s ball. So if you play a shot where the striker’s ball will end up 3 yards to the left of the aim point, the croqueted ball will move out 3 yards to the right.

In the case of a split, the same thing happens, but in a more pronounced way. The same point is sometimes put in another way: that the mid-point between the striker’s ball’s resting point and the croqueted ball’s resting point will lie on the aim line.

In the case of a roll – whether a full roll or a pass roll - the extent to which the striker’s ball will end up to the right of the aim line is the same as the extent to which the croqueted ball will end up to the left. In a sense, this is the easy one to get hold of conceptually, since the one can “halve the angle”, ie aim half way between where the two balls need to end up. This is less true for a split, and not true at all for a take off; see the picture above, which shows that the aim line is by no means half way between the direction of the croqueted ball and the striker’s ball. There is a good article of this on Ian Plummer’s Oxford Croquet site here under the heading “Aiming in Croquet Strokes - Not Half the Angle!”

Most people can feel this sideways momentum point instinctively, except the fat take-off bit, which is worth trying out on the lawn. It really works!