2o8 



NATURE 



[April 14, 192 1 



in.', and 

 A varia- 



dominate and pull the ball more quickly down to 

 earth, with resulting diminution of range. For 

 every velocity of projection of the ball which 

 leaves the tee in a horizontal direction there will 

 be a best value of underspin enabling it to attain 

 the greatest range in still air. The art of the 

 golfer is to manipulate his club so as to give this 

 necessary amount of underspin. 



It is probably not realised by many efficient 

 golfers how much this underspin may be varied by 

 small changes in the position of the line of stroke 

 of the club as it hits the ball. Let us take Tait's 

 maximum estimate of 120 revolutions or abdut 

 750 radians per second as the value of the under- 

 spin, and consider how far below the centre of 

 mass of the ball the line of impulse must be so 

 as to send the ball off with this spin and a speed 

 of 300 ft. per second. The ball is supposed to be 

 hit horizontally off the tee without any reactionary 

 upward or backward impulse acting on it. The 

 distance x below the centre of mass at which the 

 line of impulse must act so as to give this com- 

 bination of linear speed and spin has the value 

 x = W'iiilv^ where fe is the radius of gyra- 

 tion of the ball, and n and to are the speed and 

 spin respectively. With fe2 = 0-276 

 ^ = 3600 in. /sec, we find :x; = oos4 in. 

 tion of one-hundredth of an inch in this value will 

 change the spin by nearly 20 per cent. Such varia- 

 tions may easily be effected by very slight changes 

 in the lie of the club head. 



With a given ball the velocity of projection and 

 the spin are the only factors which are under the 

 control of the player. Once the short time of 

 impact between the club face and the ball is com- 

 pleted, nothing the player can do can influence 

 the flight of the ball. Thereafter all is determined 

 by the combined influence of gravity and the air. 



So far as the player is concerned, the velocity 

 of projection depends mainly upon the velocity of 

 the club at the moment it strikes the ball. The 

 weight behind the stroke no doubt has a secondary 

 influence, but the great thing is the swiftness of 

 the stroke. For this reason experience has evolved 

 a weight of club which is found most serviceable 

 for the strength of the average man. In an 

 ordinary driver, weighing (say) i lb., prob- 

 ably one-third of the weight is in the club head ; 

 and if we were to think of the problem as one of 

 simple impact between two masses of which one 

 is at rest, we might work out the relative velo- 

 cities of club and ball after impact for an assumed 

 value of the coefficient of restitution. But the 

 conditions of the problem are not so simple. The 

 player, by the swing of his body and arms and 

 well-timed effective wrist play, not only imparts a 

 rapid acceleration to the club, head up to the 

 moment of impact, but in all probability imparts, 

 unconsciously, perhaps, but none the less effec- 

 tively, an acceleration during the time of impact, 

 short though that be. In spite of the back impulse 

 on the club as it is striking the ball, its velocity 

 is kept up by the unconscious knack of the player. 

 The relative velocity with which the ball leaves 

 the club is e times the momentary velocity of the 

 NO. 2685, VOL. 107] 



club, where e is the coefficient of restitution, and 

 hence the velocity of projection will be [i + e) 

 times the velocity with which the club is moving 

 at the instant club and ball separate. 



Outside the factors over which the player has 

 some control, the most important is the resiUence 

 of the ball, and the steady improvement in this 

 quality is, of course, at the .root of the great 

 increase in lengths of drive. It was this question 

 of resilience which, indeed, started Tait on his 

 investigations on impact. The apparatus designed 

 by him for the purpose was nicknamed the 

 "guillotine." It consisted fundamentally of a 

 weight which, guided by upright parallel slots, 

 was dropped on the ball or other body the elastic 

 properties of which were under investigation. The 

 heights reached by the weight after successive 

 rebounds were recorded automatically on a rotat- 

 ing disc 2J ft. in diameter. From the record all 

 the facts of the impact could be derived more or 

 less directly, such as the compression of the ball, 

 the duration of the impact, and the value of e, 

 the coefficient of restitution. The weight was 

 made of wood, but its lower face could be, when 

 required, shod with an iron plate. 



The recording part of the apparatus has long 

 been dismantled, but the "guillotine" part is still 

 serviceable. In order to compare the values of e 

 for modern golf balls with the values obtained 

 thirty years ago by Tait, impact experiments were 

 recently carried out on sixteen balls of recognised 

 merit — namely, various types of Avon ball. Chal- 

 lenger, Clincher Cross, Dunlop, Silver King, and 

 Spalding. Thanks are due to the Avon India 

 Rubber Co., Ltd., J. P. Cochran, Ltd., North 

 British Rubber Co., Ltd., Dunlop Rubber Co., 

 Ltd., and A. G. Spalding and Bros., Ltd., 

 for their kindness in supplying specimens 

 of balls of the best quality. With the ex- 

 ception of five, all were of greater dia- 

 meter than the new standard minimum, and only 

 two exceeded the maximum standard weight. 

 Their specific gravities varied from 1-07 to 1-29. 

 On each of these balls the weight of 4-75 lb. 

 was allowed to fall from a height of 9 ft., and 

 the height of the first rebound was noted. The 

 square root of the ratio of these heights gave an 

 approximate value for e, and this was corrected 

 by comparison with Tait's results, which showed 

 that under the conditions of the experiment the 

 ratio of the speeds immediately after and imme- 

 diately before the impact was greater than the 

 estimate from the corresponding heights by about 

 one-ninth. The average value of e for the sixteen 

 balls mentioned was 072, the lowest being 0-71, 

 and the highest 075. Tait obtained for the balls 

 he experimented with the value 066. He esti- 

 mated 300 ft. per second as a fairly probable value 

 for the velocity of projection. On the assumptions 

 indicated above, this would imply a velocity of 

 projection of 311 ft. /sec. for the ball with co- 

 efficient of restitution equal to 072. 



This does not seem to indicate any very marked 

 superiority in the modern ball — at least, it cannot 

 explain the greatly increased length of drive 



