422 



NATURE 



[August 28, 1890 



for a very good drive. As the length of the path is some- 

 where about 600 feet, the initial speed must be about 

 468. This, also, is an exceedingly rough estimate, as 

 the effect of gravity has been omitted. The percentage 

 error here is the same as that of the observed time, but 

 has the opposite sign. 



Taking them together, these two estimates appear to 

 indicate an initial speed of about 500. Let us for a 

 moment assume this to be the true value, and see how it 

 will agree with the other facts of the case. 



Introducing the assumed data, we have for the typical 

 trajectory 



y = 0-258:1- - 2-524(e''/'4o _ i). 



The value of x for the maximum of y is given by 



0-258 



140 



so that X(^ = 372, and Jo = 62, at the highest point of the 

 trajectory. These values, especially that of y^, agree 

 very well indeed with those independently observed ; so 

 that we have a first hint that our assumptions cannot 

 be much in error. 



The range (so far as this approximation goes) is to be 

 found by putting y = o in the general equation. This 

 leads to 



I4'3i 



140 



(e^4o _ I). 



By the aid of a table of values of the function (e- — l)/z, 

 which I constructed for the purpose of this inquiry, I find 

 easily 



X = 140 X 4*o8 nearly = 571. 



This, again, is a tolerable approximation to the observed 

 range ; and, as above stated, we could not expect more. 

 Now nothing in golf is more striking than the well- 

 known fact that, once a player is able to drive a fairly 

 long ball, he secures comparatively little increase in his 

 range by even a great additional exertion. Assuming 

 that the additional effort is well and truly applied (and 

 this is usually, as most men too well know, a very large 

 assumption indeed) its only effect must be to increase the 

 initial speed. Let us see how an increase of initial 

 speed to 600 feet per second will increase the range, other 

 things being the same. Performing the calculations as 

 before, the rough equation for the range becomes 



20-165 = i^(6 



1); 



and A- is found to be 140 X 451, nearly, = 631 feet, or 

 only about 20 yards more. Yet the initial energy of the 

 ball was 44 per cent, greater. So far as this point is 

 concerned, our result is in good accord with experience. 

 On the other hand, if we assume the initial speed to be 

 400 feet per second only, we find 



■^= 3'55 X 140, nearly, = 497. 



This represents a fair, but not an exceptionally good, 

 drive. It thus appears that our assumption, of an initial 

 speed of about 500, meets adequately the requirements of 

 the data for a really fine drive, so far as yet tested. 



The ranges for initial speeds of 100, 200, , 600 



feet per second are, in order, 112, 277, 400, 497, 571, 631. 

 (Had there been no resistance, the ranges would have been 

 as the square numbers, i, 4, 9, .... , 36.) From these 

 data it would appear that the great majority of golfers 

 give the ball an initial speed of some 200 to 250 feet per 

 second, only : — very frequently not so much, even off the 

 tee : — and that to obtain a carry of double amount, the 

 ball must have nearly quadruple energy. 



NO. 1087, VOL. 42] 



We may now apply the test supplied by the datum (4). 

 We have, for initial speed 500, 



:^o = 372,x= 571, 



so that, in the figure above, 



AC : AB : : 372 : 571. 



The ratio is rather /ess than 2:3; whereas according 

 to observation, it ought to have been greater ; though, of 

 course, always less than 3 : 4. But I do not attach much 

 importance to this discrepancy, as the estimate made of 

 the highest point of the path is at best a rude one, and 

 depends very much upon the position of the observer. 

 For instance, it is almost impossible for him to make 

 even a guess at its true position if it should happen to be 

 situated nearly above his head. 



I have calculated a number of trajectories for larger 

 values of a, and with V correspondingly reduced, so as to 

 keep the carry the same. But all seem to give too great 

 a value for the maximum height attained ; and to place 

 that maximum too near the middle of the carry ; to suit 

 the long, raking, drives which have furnished my data. 

 The estimated value, 500 feet per second, of the initial 

 speed in "tall" drives like these, may appear a little 

 startling at first. But anyone who knows how to a^i a 

 tough ragweed with a thin cane, instead of merely 

 bruising it, as ninety-nine men in a hundred would 

 certainly do at the first attempt, will recognize the sort 

 of nip which a really skilled golfer gives at the instant of 

 striking the ball. 



It is curious to reflect that it is the resistance of the air, 

 alone, which makes it possible for the legislature to 

 tolerate the game of golf. For the normal drive which 

 was studied above would, but for the resistance of the air, 

 have a carry of 1250 yards (more than two-thirds of a 

 mile) and the ball would fall at that distance with its full 

 initial speed of 500 feet per second ! The golfer might 

 deal death to victims whom he could not warn with the 

 most Stentorian "Fore," He could carry, at St. Andrews, 

 from the first tee to the " Ginger Beer " hole ! This illus- 

 trates, though in a very homely and feeble way, the ser- 

 vice which the atmosphere is perpetually rendering us by 

 converting into heat the tremendous energy of the in- 

 numerable fragments of comets and meteorites which 

 assail the earth from every side with planetary speeds. 



When there is a steady wind, even when it blows in the 

 plane of flight, the mathematical problem is much more 

 difficult : — and this difficulty is not sensibly less when an 

 approximate solution only is sought. For the speed of 

 the wind depends on the height above the earth ; and, 

 even if we take the simplest law for this dependence, 

 neither of the equations can be treated separately. 



It is easy, however, to see the general nature of the 

 effect. In driving against the wind, the resistance (which 

 of course depends on the relative velocity) is greater than 

 in still air : — but its direction is no longer in the line of 

 flight, except at the highest point of the path. It acts in 

 a direction less inclined to the horizon than is the path, 

 and therefore its effect on the horizontal component of 

 the velocity, as compared with that on the vertical com- 

 ponent, is greater than in still air. With a following 

 wind, unless it be going faster than the ball, the opposite 

 effects are produced. The general result is to affect the 

 carry considerably, and the vertical motion but slightly. 

 The time of flight is probably a little shortened by a 

 following wind, while it is lengthened by a head wind. The 

 belief, prevalent among golfers, that a ball rises higher 

 against a head wind, and lower with a following wind, 

 than it would do in a calm, is due directly to the effect of 

 perspective :— the highest point of the path being shifted 

 nearer to, or further from, the player. The true effects on 

 the greatest height reached are usually too small to be 

 detected by a casual observer. 



