The Academy of Quantitative Billiards
In the last issue we invited answers to this problem:-
The table at our club is a fast one: a ball hit at top strength will travel six lengths (i.e. it will double the table lengthwise three times). The length of the table, measured from the nose of the top cushion to that of the bottom cushion is exactly 12ft. If the cushion is depressed by the ball's impact 1/4 inch on average, what is the total distance traversed by the ball if it comes to rest exactly at its starting position after a six length bash?
The answer is more than one foot less than many people would think. Including the effect of cushion distortion, a single lengthwise traverse from the bottom to the top cushion, measured with a tape, comes to 12ft plus one half inch. But, the centre of the ball can never get closer than one half a ball width to the cushion nose; therefore, the centre only traverses the above distance less one whole ball width. The answer to the problem is thus 72ft. 3in. less six ball widths. And we did say that his width is 2 1/16 inches!
How Fast Does the Cue Ball Travel
The distance of seventy-odd feet just established does not seem to be especially impressive for a full-power shot: the equivalent shot in golf is, after all, about ten times as long. This difference originates (a) in the much higher initial speed of the golf ball and (b) in that the golf ball encounters only air resistance up to the time when it first hits the ground, several seconds after being hit. The billiards ball, on the other hand, has its kinetic energy of motion greatly reduced by its impact at speed with the cushions, the first of which occurs a fraction of a second after striking the ball. The cushion tends to act as a brake on the rotational movement of the ball during the time of its impact. The question naturally arises as to how far a ball would travel in the absence of such energy-destroying impacts. To answer this question it is necessary to know how fast a hard-hit billiards ball moves just after it has been hit by the cue tip. This speed can be calculated from successive frames of a cine film of the shot (as Riso Levi recognised in 1930) or from a video, but nobody seems to have made the measurement with an English billiards ball. In the U.S.A., George Onoda has made the calculations made from video recordings of the break shots in US pool, and finds this speed is about 25 mph. This seems disappointingly slow!
One can calculate the speed at which one hits the ball oneself using just the kitchen table. One simply hits the ball off the table edge and measures the horizontal distance D it travels by its first bounce on the floor. If the table height is H, H and D being measured in feet, then the initial speed of the ball is given by the expression 4D/ H ft. per second, or (nearly) 3D/ H mph. D should be about 12ft, but please note that neither the authors nor the BQR will underwrite the cost of house repairs after this experiment in Quantitative Billiards! Note
that the ball is assumed to be flying horizontally when it leaves the edge of the table. If it is actually struck above the centre with a downwards blow, it will in fact describe a low trajectory over the table surface and this would invalidate the calculation just given. The calculation applies without change to the problem of calculating the range of a cannon ball fired out to sea from a cliff-top and it was probably first performed with this end in view. At all events, we are not claiming to be the patentees of the above formula!
Ball fired off table with speed V (V=4*D/H)
More about the problem of the ball hit up a very long table in a later issue. Those fortunate enough to possess Riso Levi's, "Billiards in the Twentieth Century," will know that he had thoughts of his own on this topic, although he approached the problem in another way.