THE EFFECTS OF SLOPING COURTS
evident to the striker, but it shows up clearly when there is dew on the court. Were it not for the dampening effect of fric7on, the ball would actually spiral around its des7na7on before coming to rest. In prac7ce, the most we ever see of this final phase is what can appear to be the ball lurching to one side. This is the last gasp of the precessionary mo7on, not (as is some7mes suggested) evidence of a bias in the ball. The increasing curvature in the ball’s path means that, roughly speaking, it will only have deviated by about 15% of its eventual dri' down the slope when it reaches the halfway point of its journey. Hi9ng the ball harder than is necessary to reach a target therefore makes a lot of sense if the distance it travels does not ma8er: it ensures that the object ball is reached before the worst of the curvature comes into play. But hi9ng the ball harder tends to reduce the accuracy of the shot, so there is a balance to be struck – and for some shots, it will also ma8er how hard the target ball is hit. If the ball is hit with the same strength on a slower court, it will deviate rather more – at a given distance – simply because it needs to travel propor7onately further along its curved path. But the differences are not large within the normal range of court speeds. Calcula7ons show that even a small slope of 2mm per metre (about 50mm over the full width of a court) is enough to cause problems for shots at distant targets. On a 12 second court, for example, a ‘perfect’ shot hit with sufficient strength to travel about 55m will miss any ball which is more than about 24m away; with a 5mm per metre slope and a shot of the same strength, this cri7cal distance falls to about 15m – not much more than the distance of a short li' shot. Incidentally, for those of us s7ll not fully metricated, these modest slopes translate neatly into inches drop across the width of a court (a full‐sized court is 1008 inches wide – and covers an area of just under 1000 square yards). In prac7ce, of course, shots are not consistently accurate. Aiming or striking errors will push the shot away from its intended target, some7mes onto the path which the ball actually needs to follow to compensate for the slope. So how much effect on the outcome does the slope have in prac7ce? And does a slope really favour less accurate players, who may rely rather more on ‘lucky accidents’ than skill to hit long shots? The answer depends on both the extent and the varia7on of these errors. A useful baseline for accuracy is provided by the comprehensive sta7s7cs gathered by Dr. Grundy, mainly in the 1930s, which were summarised by David Prichard in the Summer 1977 issue of the Gaze8e. This gives us a measure of the average accuracy of tournament players over a range of distances. Dr. Grundy’s data only tell us about the hit rate, not the sca8er of shots away from the centre of their target. But his data are a good fit to a normal (bell curve) distribu7on whose standard devia7on increases pro rata with distance, which is a convenient star7ng point for modelling shoo7ng accuracy.
By Ian Bond
A formal exposi0on of the physics and mathema0cs underlying this ar0cle can be found in „A. R. Penner, The physics of pu2ng“, Canadian Journal of Physics (2002) or „Rod Cross, The trajectory of a ball in lawn bowls“, American Journal of Physics (1998), both of which are available online at no cost. Lacking the knowledge to devise the relevant equa0ons myself, I have simply followed their recipes – with parameters calibrated for croquet balls and courts – to draw some conclusions relevant to croquet. As we all know, croquet courts are rarely perfectly level. But how much difference does this make to our success rate at hi9ng or reaching our targets? Is it be8er to play against a good shot on a poor court, on the assump7on that this will weaken the advantage they gain from their accuracy? And how close to perfectly level do our courts need to be, to make the effects of any overall slope irrelevant for all prac7cal purposes? The first ques7on to deal with is extent to which a slope will affect the accuracy of a shot across it. This turns out not to be straigh6orward. Ins7nc7vely, one might suppose that it is simply gravity and the slope of the court which take the ball off course. But a moment’s reflec7on tells us that that cannot be the whole story. The ball does not roll down the slope before the shot is taken, nor does the slope prevent it from coming to rest once it is in mo7on. So both sta7c and rolling fric7on must be enough to stop the ball running down the slope, away from the intended line of the shot. It is only on quite severe slopes – rarely encountered on a decent croquet court – that gravity by itself will be able to do the job. Instead, the key driver of the ball’s curved path across a sloping court is its rolling mo7on once it has been struck by the mallet. Because the ball’s centre of mass is not directly above its point of contact with the court, its rolling generates a second rota7on around that main axis – a precessionary mo7on. This takes the ball away from its intended course and con7nues only for as long as the ball is rolling across the court. The earth’s axis of rota7on also precesses, very slowly, under the gravita7onal influence of the sun and moon. An important prac7cal result is that the ball follows a curved path, which becomes more 7ghtly curved as it con7nues on its journey: the further it travels, the more its trajectory is angled away from the intended line. This is not always
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