Dr Ian Plummer

http://www.oxfordcroquet.com/tech/kibble1/index.asp

Oxford Croquet

David Kibble investigates the aiming accuracy needed to produce a successful cut rush in a desired direction. He demonstrates that for small cut angles the aiming accuracy is relaxed (!) however for big cut angles more accuracy is required.

As many before me, I discovered what a marvel of human ability it is to roquet a ball more than a couple of feet away. With almost no distance between the balls the target angle is 180°, at a yard it’s 11°, two yards is 5°, at just over 11 yards it's one degree. A tea lady presents a target of about one third of a degree – that’s less than the moon!

So to a rush and Martin Murray’s assertion: "I remember being told years ago that if the balls are less than a ball's diameter apart, the rush is more accurate than a single ball stroke".

The striking angle is amplified (or reduced) by the distance between the balls in a rush. With two balls almost in contact, striker’s ball cannot "see" much of the target ball’s surface – whichever direction striker’s ball travels (as long as it hits the target), it will strike more or less the same point on the ball, so the ball always goes off in the same direction. As the initial gap between the balls increases, it becomes possible to hit more of the target ball, thus possible to send it off in different directions – note that to cut a ball at 90° you have to start from a very great distance! The target the ball presents very much reduces with distance – the sensitivity of the cut direction to the striking direction increases with both distance and the amount you are trying to cut it (i.e. the bigger the cut, the more accurate you have to be) – a good argument for rush-line theory perhaps.

I imagine a pair of scissors with the pivot at the centre of the ball that’s being rushed and I move the handles on the surface of the ball to the extremes at which the ball can be hit by a striker’s ball. Note that in close proximity, most of the ball is hidden by the near parts - just like when standing on a mountain you can only see to the horizon, and that’s not much of the globe. The scissor blades point to the extremes at which the ball may be cut. With the balls in contact, the scissors are closed (there is no variation, whichever direction the ball is struck) and they gradually open up with increasing distance as more of the ball becomes hittable.

Of course, all this assumes perfect collisions between perfect spheres, etc.

At a separation of half-a-ball the cut angle is around half the striking angle right up to a striking angle of 15° where it starts to increase, and it passes unity at about 30° (i.e. 30° striking angle = 30° cut) – a 42° striking angle misses the ball.

At a separation of one-ball the cut angle and striking angles are the same up to around 10°, the sensitivity becomes two at around 20° (one degree of change in striking angle causes two degrees of change in cut) – 30° misses the ball.

Enough of that, and on to the original question using rush lengths most normal players are faced with. The equivalents are:

In English...

A one-foot rush is missed at 13.5° and has a sensitivity starting around 3.5 that starts to rise rapidly at 30° of cut (<10° striking angle). To send it straight (±5°) needs a striking angle of ±1.5° (equivalent accuracy to hitting a 4-yarder). To cut it at 45° (±5°) needs a striking angle of 10.3 to 11.8° (equivalent to hitting an eight-yarder).

A two-foot rush misses at 7.5° and has a sensitivity starting around 7 that starts to rise rapidly at 30° of cut (<5° striking angle). To send it straight (±5°) needs a striking angle of ±0.7° (equivalent to hitting an eight-yarder). To cut it at 45° (±5°) needs a striking angle of 5.4 to 6.2° (tolerance 0.8°, equivalent to hitting a 14-yarder).

A one-yard rush misses at 5.2° and has a sensitivity starting at 10 that starts to rise rapidly at 30° of cut (<3° striking angle). To send it straight (±5°) needs a striking angle of ±0.5° (equivalent to hitting an eleven-yarder). To cut it at 45° (±5°) needs a striking angle of 3.6 to 4.2° (equivalent to hitting a 19-yarder).

A two-yard rush misses at 2.8° and has a sensitivity starting at 20 that starts to rise rapidly at around 25° of cut (<1° striking angle). To send it straight (±5°) needs a striking angle of ±0.2° (equivalent to hitting a 29-yarder). To cut it at 45° (±5°) needs a striking angle of 1.8 to 2.1° (equivalent to hitting a 39-yarder).

The lessons are: get close and get on the rush line – no surprises there then :)

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It was suggested that it would be helpful if these angles could be converted into 'equivalent to hitting a ball X yards away'. The table below indicates the half angle (i.e. left and right of the aiming line) through which the mallet can be twisted and still hit a ball X yards away.