September 24, 1891] 



NA TURE 



497 



SOME POINTS IN THE PHYSICS OF GOLF. 

 II. 



IN my former paper {Nature, Aug. 28, 1 890) the main 

 conclusions were based to a great extent upon the 

 results of mere eye observations, often of a very uncertain 

 and puzzlmg kind. The data so obtained were unfor- 

 tunately not those required for a direct investigation, so 

 that my processes were necessarily of a tentative character. 

 During and since the last College session I have been 

 endeavouring to obtain some of the more important data 

 in a direct manner. I am thus in a somewhat more 

 favourable position than before but, as will soon appear, 

 the new information I have obtained has complicated 

 rather than simplified the singular problem of the flight 

 of a golf-ball. 



One point, however, which is both curious and important, 

 has been clearly made out -.—havimering has no effect (or, 

 to speak more correctly, only an inconsiderable effect) on 

 the coefficient of restitutiott of a golf ball. This conclusion, 

 which may have to be modified if the striking surface be 

 not plane, had for some time appeared to me as almost 

 certainly correct, and I have recently verified it by means 

 of the Impact apparatus with which I have been working 

 for some years. I procured from St. Andrews a number 

 of balls of the same material and make, half of them only 

 being hammered, the others plain. The results obtained 

 from a hammered, and from an unhammered, ball did not 

 differ much more from one another than did those of a 

 number of successive impacts on one and the same ball. 

 [In the Badminton Libraiy\o\\.\mQ on Golf, Mr. Hutchin- 

 son quotes a statement of mine which appears at first 

 sight diametrically opposed to this experimental result ; 

 and thus puts me in the position de nier ce qui est et 

 (fex-piiquer ce qui n'est pas. But he has omitted to men- 

 tion that my statement was expressly based on the alle- 

 gation that a hammered ball had been definitely found 

 to acquire greater speed than an unhammered one. This 

 seemed to me even at the time very doubtful, and I now 

 know that it is incorrect.] Thus it is clear that the un- 

 doubtedly beneficial effects of hammering must be ex- 

 plained in some totally different way. There is another, 

 and even more direct, mode of arriving at the same con- 

 clusion. To this I proceed, but unfortunately the new 

 point of view introduces difficulties in comparison with 

 which all that has hitherto been attempted is mere child's 

 play. In short, it will be seen that the problem of a golf- 

 ball's flight is one of very serious difficulty. 



In my former article 1 took no account of the rotation 

 of the ball, treating the problem in fact as a case of the 

 motion of a particle in a medium resisting as the square 

 of the speed. The solution I then gave was only ap- 

 proximate, and limited by the assumption that the cosine 

 of the inclination of the path to the horizon might be 

 treated as unity throughout. The illustrations and ex- 

 tensions given were founded on the same basis as was 

 the solution of the simpler problem. -Shortly after it was 

 published I made, by the help of Bashforth's tables, a 

 more exact determination. The data I thus arrived at 

 were (in Bashforth's notation) 



^ = i'9, «o = *3i feel-seconds, (p = I3°'5- 



From these the tables give at once 



Range of Carry = 542 feet 



Maximum Height ... ... ... = 5^ >» 



Horizontal Distance of Highest Point 



from Tee = 350 ,, 



Initial Speed = 480 feet-seconds 



Terminal ,, ... = 80 ,, 



Terminal Inclination = 38°'5. 



As a contrast, take X = ri, so that Uq = 100 feet- 

 seconds. To obtain the observed range we must take 



NO. I 143, VOL. 44] 



({) = 23°-25, which is considerably too great. The other 

 numbers then become 



Range of Carry = 543 feet 



Maximum Height = loo ,, 



Horizontal Distance of Highest Point 



from Tee = 35° .. 



Initial Speed = 393 feet-seconds 



Terminal ,, = 80 ,, 



Terminal Inclination — 54°'6 



The first numbers are in remarkable accordance with the 

 numerical details of really good drives which I obtained 

 from Mr. Hodge ; and, were there no other crucial test 

 to be satisfied, the problem might have been regarded as 

 solved 10 at least a first approximation. But I felt very 

 suspicious of the sufficiency of such a solution ; espe- 

 cially as it made no place (as it were) for the possibility 

 of a path in part straight, or even occasionally concave 

 upwards, which I have certainly seen in many of the 

 very best drives. And my doubts were fully justified 

 when I calculated from Bashforth's tables the time of 

 flight under the above conditions. For they give 1-515. 

 for the first, and 2-i3s. for the second, part of the path : — 

 3'6 seconds in all ; while the observed time of flight in a 

 really good drive is always over 6 seconds, and some- 

 times quite as much as 7. This I have recently verified 

 for myself with great care in the competition for the 

 Victoria Jubilee Cup, where one of the unsuccessful 

 players distinguished himself by really magnificent 

 driving. The time of flight in the second of the above 

 forms of path is about 48 seconds. 



The initial speed in the first estimate seems to be 

 excessive, as will appear from the experiments to be de- 

 scribed below. This, of course, is one mode of explain- 

 ing how the time of flight is so much underrated. But, 

 if we keep to Bashforth's value of the coefficient of re- 

 sistance, it is impossible to reduce the initial speed (while 

 preserving the observed range) without increasing the 

 angle of projection and, with it, the greatest height 

 reached. The second set of numbers conclusively 

 proves this. On the other hand if, with the view of re- 

 ducing the initial speed and thus increasing the time of 

 flight, we assume a smaller resistance, we may keep 

 range, height, and initial angle, nearly as observed ; but 

 we shift the vertex of the path unduly towards the mid- 

 range. The only way, it would therefore seem, of recon- 

 ciling the results of calculation with the observed data, 

 is to assume that for some reason the effects of gravity 

 are at least partially counteracted. This, in still air, can 

 only be a rotation due to undercutting. 



During last winter I made a considerable number of 

 experiments with the view of determining the initial speed 

 by the help of a ballistic pendulum, but the results of 

 these cannot be regarded as very satisfactory. My 

 pendulum was a species of stiff but light lattice-girder 

 constructed of thin, broadish, laths. This hung from hard 

 steel knife-edges set well apart, and supported a mass of 

 moist clay of about 100 lbs. The clay was plastered into 

 a nearly cubical wooden frame, and swung just clear of 

 the floor. The ball was driven into it from a distance of 

 about six feet, and as near as possible to the centre of 

 one face. The effective length of the corresponding 

 simple pendulum was about 10 feet, and the utmost de- 

 flection obtained (measured on the floor) was about two 

 inches. From these data I deduced an initial speed of 

 about 300 feet per second only. But the experiments 

 were never quite satisfactory, as the player (however 

 skilful) could not free himself entirely from appre- 

 hension of the consequences of an ill-directed drive. 

 In fact, several rather unpleasant accidents occurred 

 during the trials, especially in the earlier stages ; when 

 the pendulum was mounted in a stone cellar, and without 

 the hangings and the paddings which were employed in 

 the later work. Although the clay was so stiff as to 



