Technical
Where a Ball Ends Up after a Collision
Simple TheoryIf a moving ball collides with a stationary one, then the moving ball should be replaced along its original trajectory at the point corresponding to the sum of the distances travelled by both balls from the point of collision. The above result incorporates the assumptions that there is a constant deceleration of the ball^{1}, the collision is elastic (energy is conserved) and the balls have the same weight. A simple proof of why the distance travelled by a ball is proportional to its energy is given in the Appendix. Under those conditions the distance travelled by a ball is directly proportional to the energy it carries. This is reassuring – hit the ball with twice the energy and it goes twice as far. When the balls collide the energy is shared between them and they each move a distance corresponding to the fraction of the energy they received. Add the distances travelled by the separate balls and it gives you the distance which would have been travelled by the unhindered ball. Energy LossesIt has been noted by Max Hooper who used a malletswinging rig, that the method above underestimates the final position of the ball. This is not surprising given the assumptions. The assumption that the collision of the ball is elastic (energy is conserved on collision) is patently false. The coefficient of restitution (bouncyness) of a croquet ball is defined in the Croquet Laws to lie between 5075% (6th Edition Laws Appendix 2; a ball must bounce between 3045" when dropped from 60" on to a defined rigid surface). Consequently 2550% of the energy is lost during a headon collision. We know that distance scales with energy hence we should need to multiply the sum of the scattered distances by up to 2 times. The amount of energy available to be transfered between balls during a collision also depends on the geometry. There is no real problem with the assumptions that both balls weight the same. The final assumption that the balls undergo a constant deceleration (which justifies the 'distance scales with energy') is a rough approximation. A curve fitting was used on experimental data to produce a relationship^{1}. The data is sparse for when the ball is moving slowly (i.e. for short total travel distances = short transit times). It may be that a slowly moving ball decelerates less than a quickly moving ball. Further measurements are needed but the empirical data indicates that the estimate of constant deceleration is good and not likely to have as large an effect on the Simple Theory as the assumption that energy is conserved. There are additional factors which affect the forces on the balls after collision, some may be nonlinear with ball velocity. These include:
All of these will act as 'energy leaks' and hence may make the deceleration dependent on velocity (hence not constant). It is, for example, well known that air resistance scales as the square of velocity. More experimentation is needed. An Australian SystemA system is proposed in the Australian Croquet Association (ACA) Referees Manual (2003, Page B16). I hope I have paraphrased their solution accurately below (thanks to Owen Edwards for the recipe). As indicated in the caption the Red ball is replaced the distance it was scattered from the point of collision plus two and a half time the distance the Brown ball was scattered. From the above sections in this document it is clear that the multiplication factor (2.5) is an attempt to scale for the energy losses on collision and for what happens subsequently. For certain collision configurations this is plausible. Consider however the two circumstance below: A) where Red hits Brown almost full on  as in a rush, and B) where Red gives Brown only a slight glancing blow. The situtations are approximatey symmetrical with the balls interchanged. Case A) is the hardest to reconcile. In the rushlike situation, Brown is boosted almost as far as Red might have travelled. Yet Red gets replaced two and a half times y_{1} up the lawn. Given the approximate diagrams above it seems unlikely that x_{1} + (2.5 * y_{1}) = x_{2}+ (2.5 * y_{2}). Experiments have yet to be repeated. Appendix  Distance Travelled is Linearly Proportional to EnergyThe almost mathematicsfree version is: Energy = force x distance, the force being the resistance of the grass (constant, by assumption). If you start two balls with say half the energy, they each will go half the distance; add the two distances together and it is the same as a single ball with all the energy. We can attempt to make it appear more thorough by using Newton’s Equations of Motion:
where
Consider a ball rolling along the grass; it will stop when its velocity becomes zero after a time t_{f} – the ‘final’ time. Putting v = 0 into Eqn. 1 and rearranging;
substitute t= t_{f} in Eqn. 2 and we get the total distance, s_{f}, travelled by the ball:
A little bit of deviousness now in the calculation. We use the standard equation for the initial kinetic energy:
This is the simplification; the ball will hold its energy in both kinetic and rotational energy but those forms are readily interconvertible. We would be surprised if a ball came to rest but was still spinning! We can rearrange Eqn. 5 to get u, the initial velocity, in terms of E:
We can now substitute for u^{2} in Eqn. 4
This result shows that the total distance, s_{f}, is a simple multiple of the initial energy, E. Note that the 'acceleration', a, is negative designating a deceleration due to the friction, etc. against the grass and ground. This yields a positive distance travelled. Another way to phrase it is that the energy decreases linearly with distance. Footnotes1. Lawn Speeds: http://www.oxfordcroquet.com/tech/lawnspeed/index.asp All rights reserved © 20082014
