In search of the perfect mallet ©
Robert Fenwick Elliott 20052006

Basic Theory of Dynamics
There are a
number of questions, such as
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 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 NorthSouth
axis, and separate calculations for the motions along the EastWest axis. 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:
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, 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 northsouth,
but will not matter if it is anything else).
Immediately before impact, the mallet will have lots of momentum
northsouth, but none eastwest. 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 eastwest momentum is nil.
The total eastwest 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:
Most people can
feel this sideways momentum point instinctively, except the fat takeoff bit,
which is worth trying out on the lawn.
It really works! 