Dr Ian Plummer
"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."
More precisely straight rushes can be calculated to be more accurate in certain circumstances as Nick Furse demonstrates below.
The following diagrams show three rushes.
S is the centre of the striker's ball, H is its position when it just contacts the rushed ball (centre R).. T is the target and F the final track of the rushed ball.
The angle α is the angle that the mallet is swung off-line and β is the angle that the rushed ball departs.
In diagram 1: α = β; diagram 2: α < β and diagram 3: α > β
In the first, the striker's ball moves one ball length and the accuracy of the rush is virtually the same as the striker's ball. In the second, the striker's ball has to travel further and the rush is less accurate. Finally, in the third, when the striker's ball travels less than a ball's width, the rush is more accurate.
For a rough 'rule of thumb', the further the striker's ball travels before striking the rushed ball, the more accurate the stoke must be to hit the target..
For very short rushes (less than a ball's width) the direction in which striker's ball is struck becomes less significant and so "accuracy" will increase and tend towards that of a croquet stroke..
Mathematically:
SH * sin(α) = RH * sin(β)
Thus
sin(β) = (SH / RH) * sin(α)
Since RH = 1 ball width (centre to centre when in contact), then:
sin(β) = SH * sin(α) when SH is measured in ball widths.
Examples:
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When SH = 1 (as in the first diagram), sin(β) = sin(α) and β = α.
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In an example of a two yard rush, because the length of the rush is about 20 ball widths in length (more precisely 2*36/3.625 = 19.86), the rush is about 20 times less accurate than the equivalent single ball shot. Thus if the rush misses the target by a yard, the equivalent single ball shot would miss the target by only about 36/20 = 1.8 inches - certainly close enough to hit a ball or peg if they were the target.
Having said all this, the assumption that the rush is perfectly in line with the target is critical. This is difficult to judge when the balls are very close together. Players have no real practice at lining up such rushes (unless their opponent accidentally leave a ball a few inches from a balk line when conceding a lift).
Some may also have noticed that as the gap between the balls reduces to zero so does the inaccuracy of the rush. This assumes that a correctly lined up croquet shot will hit the target every time. Reality is not like that because human error occurs in lining up croquet shots as well as judging whether or not a rush is perfectly straight when aiming at a distant target. All this is quite apart from other factors such as pull and lawn conditions etc.
Nick.
Additional Derivation - relationship between α, β and SR in "Angles and Separations in a Rush Shot"
See also Aiming Accuracy in Cut Rushes