August 28, 1890] 



NATURE 



421 



the club. A drive was then made, in a direction parallel 

 to the axis ; first, with the disks at rest ; second, when 

 they were revolving about nine times per second. From 

 the result of the first experiment, the correction for the 

 second, due to the fact that the club did not move exactly 

 parallel to the axis, was roughly determined. The results 

 obtained varied within wide limits ; i.e. from 140 to 700 

 feet per second for the speed of the club-head at impact. 

 But the majority of the experiments gave from 200 to 

 300 feet per second. The golfer whose services I enlisted 

 for these experiments, though a very good player, con- 

 fessed that the novelty of the circumstances had pre- 

 vented his doing himself justice : — the revolution of the 

 disks, in particular, tending to prevent him from "keep- 

 ing his eye on the ball." There can be little doubt that 

 the main cause of discrepance among the results was the 

 fact that the correction had to be found when the disks 

 were at rest, and to be applied to data obtained when 

 they were moving. At the time, I formed from these 

 experiments the conclusion that the initial speed of the 

 ball must be somewhat over 400 feet per second. I have 

 since been led to believe that this is an under-estimate. 

 I hope when I return to my Laboratory, to carry out this 

 class of experiments with more satisfactory results ; by 

 repeating, under favourable conditions, an electrical 

 process which recently failed from the employment of 

 inadequate apparatus. 



So long as the speed of a spherical projectile is less 

 than that of sound, it appears that the resistance of the 

 air is at least approximately as the square of the speed. 

 (It is on this account that the effect of even a light head, 

 or following, wind is so considerable. For it is the relative 

 speed that determines the resistance, and even a small 

 change in a quantity makes an important change in its 

 square.) Our knowledge of this question is as yet very 

 imperfect ; but we cannot fall into any egregious error 

 by making our calculations on the assumption that this 

 law is correct. To apply it, however, we require a 

 numerical datum, e.g. the resistance (in terms, say, of the 

 weight of the golf-ball) for unit speed. 



Robins, more than a century ago, gave as the result of 

 experiments a statement equivalent to the following : — 

 The terminal speed of an iron sphere in ordinary air is 

 that which it would acquire by falling, in vacuo, through 

 a space of 3001^ yards, where d is the diameter in inches. 



From this it is easy to calculate that the resistance- 

 acceleration of a golf-ball should be about 



400 



where v is the speed in feet per second, and the denomi- 

 nator is 400 feet. 



In the recent edition oi\The Bashforth Chronograph'^ 

 we find that, for an iron shot whose diameter is d inches, 

 and mass w pounds, the acceleration due to the resistance 

 of the air at speed v (expressed in feet per second) is 



_ 1 18-3 </^ 7/2 



W ' lOOO^* 



It is clear that this expression holds for spheres of any 

 material. For the whole resistance depends only on the 

 size and speed, while the acceleration due to it is inversely 

 as the mass. Now for an average golf-ball d= 175 

 nearly ; and w = oioi, because the specific gravity of 

 gutta-percha is nearly the same as that of water. Hence 

 we may express the acceleration by 



- il 

 280 



very nearly : — the denominator being in feet. 



I have decided to employ Bashforth's result as probably 



' Cambridge University Press, 1890. For -this reference, and for some 

 much needed explanations, I am indebted to Prof. Greenhill. 



NO. 1087, VOL. 42] 



the more accurate :— my own independent estimate, above 

 alluded to, having given 300 in place of 280. It indicates 

 resistance some 43 per cent, greater than that deduced 

 from the older reckoning of Robins. In the formulae 

 below we will write a for Bashforth's 280 feet. 



For a golf-ball not under the influence of gravity the 

 equation of motion would therefore be 



which gives, if V be the speed when / = o, 

 I I t^ 



1 + 



V/ 



From this we have 



= '"°K'+?> 



and 



Vc 



Thus in general, as e-°'7 = ^ nearly, the speed, what- 

 ever it be, is reduced to half when the ball has moved 

 through 196 feet, or about 65 yards. The time of passage 

 is 280/V. 



In treatises on Dynamics of a Particle (Taitand Steele, 

 for instance) it is shown that, for the assumed law of re- 

 sistance, the approximate equation of a flat trajectory is 



,2 2r 



^ = (tan a .,^,).-^, (..-,). 



4Vo2 



In obtaining this result it has been assumed that dx\ds may 

 be treated as being practically unity. This gives a fair 

 approximation to the form of the path of a golf-ball up 

 to, and a little beyond, its highest point ; but can scarcely 

 be relied on for the last 30 yards or so of the path, where 

 the inclination to the horizon becomes considerable. But 

 the error will not be a very serious one. If we reject this 

 approximate equation we are forced to use the intrinsic 

 equation of the path, which can be integrated exactly. 

 But, though its use can be made comparatively simple 

 by employing a graphic method, it is always very tedious, 

 and therefore only to be resorted to in the last extremity ; 

 and when we are in possession of data far more exact 

 than any yet obtained. The same may be said, so far as 

 data are concerned, of the elaborate Tables calculated by 

 Bashforth. If we had accurate information as to the 

 speed at the highest point of the trajectory, these would 

 give us all that could be desired. 



In the above formula Vq represents the hori- 

 zontal component of the initial speed : — or, practically, 

 with the limitation introduced, the initial speed itself, 

 a is the angle of projection, and has been carefully deter- 

 mined as on the average about I3°'5. Its tangent is o'24. 

 Mr. Hodge, to whose valuable assistance I owe this as 

 well as many of my other data, found it absolutely neces- 

 sary to use a clinometer, as the eye-estimates of the angle 

 of projection are almost always greatly exaggerated. The 

 only other datum required to complete the equation is an 

 approximate value of V. Two methods of finding it were 

 tried, as follows : — 



From a number of (necessarily very rough) observa- 

 tions, made by holding to the ear a watch ticking 4 times 

 per second, it seems that in the first second a well-struck 

 ball goes on an average somewhere about 100 yards. 



Hence the initial speed must be about 



28o(e' 



= 537 feet per second. 



An error of i p.c. in this measurement entails i'6 p.c. 

 error in the result. 

 The average time of flight seems to be about 45 seconds 



