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Dr Ian Plummer

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Replacing Balls after Collisions

In a double-banked game, a ball from one game (the striker’s ball) may collide with a ball from the other game (the other ball).  The other ball has to replaced in its original position, while the position of the striker’s ball is estimated from the distances travelled by it and the other ball.

A ramp was set up and balls were run down it, first to see where the ball would finish up without a collision, then placing a ball in its path to see where each ball went after the collision.

The following were the results:

Distance without collision (m)

Striker's ball distance

Other ball distance


























* The ratio is the value of 'a' in the equation d = a * x + y where d = distance travelled without a collision; x is the distance travelled by stationary ball; and y is the distance travelled by the moving ball.

Thus, the figure of 2.5 times the distance travelled by the stationary ball, plus the distance of the moving ball,  is a reasonable estimate of the correct placement of the moving ball.

It is interesting that hitting the other ball head-on and hitting it at an angle made little difference to the ratio.

Vertical view

Side view of set-up

Top view

Plan view of collisions

Neil Hardie and ramp

The author with ramp

Appendix - Method

I did the tests when I found the 2.5 plus 1 rule in Owen Edward’s Australian referee manual, and I did not think the formula credible.  I was surprised when I discovered that it fitted actual tests very well.

The ramp sloped at a little over thirty degrees – I did not measure it carefully.  The slope was judged to be sufficient to get enough velocity on the ball, but avoid too much bouncing up and down when it reached the ground.  There was a gap of two metres between the ramp and the balls placed for collisions, as I wanted the ball to be rolling normally when the collision happened.

The speed was controlled by placing the ball at various heights up the ramp (we did a few  quick tests to establish that from a given point the distance travelled without a collision was consistent).  The tests involved measuring the distance travelled without a collision, and then the distance travelled by both balls when a collision occurred, with the ramp ball placed at the same point on the ramp each time.

There was a shower of rain in the middle of the tests, so we got to do tests on a fast and slow lawn, with no significant difference.

The tests were done on Dawson International balls.

We did not record the end positions of the balls, only the distance travelled (but it would not be too difficult to calculate the angles given the ratio of distances travelled) as our aim was simply to test the formula given.  If we had not got such a good fit, we would have explored other possible equations.

It is surprising that the ratio is as high as 2.5 – but if you look at the coefficient of restitution of the balls this becomes reasonable.  The coefficient is between 0.5 and 0.75 – say an average of 0.625.  When balls collide, you need to take both coefficients into account, so the energy transfer is .625 times .625, or 0.39.  That corresponds well with a ratio of 2.5 for distance travelled.

I am certainly not aware of hitting rushes 2.5 times as hard as take-offs, but this appears to be the case.

Author: Neil Hardie
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Updated 1.viii.17
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