420 



NATURE 



[August 28, 1890 



SOME POINTS IN THE PHYSICS OF GOLF. 



IT is not an easy matter to determine the initial speed 

 of a golf-ball : — but this is so only because the direct 

 processes which have given us so much information about 

 the flight of military projectiles are here practically in- 

 applicable. No doubt, a ballistic pendulum, or a Bash- 

 forth chronograph, might after long and tantalizing 

 experiment give us the desired information. If they did, 

 they would give it much more accurately than we are 

 otherwise likely to obtain it. But the circumstances of a 

 " drive " at cricket or golf are so uncertain, even with the 

 best of players, that it would be waste of time, and wan- 

 ton vexation of spirit, to employ these instruments of 

 precision. Yet the questions involved are of a very in- 

 teresting kind, not only from the purely physical point of 

 view but also in consequence of the recent immense 

 development of these national games ; so that there is 

 considerable inducement to attempt at least a rough 

 solution of some of them. 



The following investigation, because based mainly on 

 mere eye-observations usually of a rather uncertain and 

 difficult kind, is offered only as a rude attempt at a first 

 approximation ; and I am quite prepared to find myself 

 obliged to modify the results, when new and more accu- 

 rate information is forthcoming. 



My main reason for bringing it forward in such a con- 

 dition is to enlist if possible (at this, the proper season) a 

 few keen and accurate observers, who may occasionally 

 find themselves in a position to obtain data of real value. 

 Thus I shall devote what might otherwise be considered 

 an excessive amount of attention to the nature of the 

 real desiderata, and to the quality and the sources of the 

 more common errors of the estimates which have been 

 kindly furnished to me. Such as they were, however, 

 they enabled me to state to the Royal Society of Edinburgh, 

 on July 20, conclusions as to initial speed, and coefficient 

 of resistance, nearly agreeing with those given below. 



The influence of even a moderate wind on the flight of 

 a golf-ball is so very considerable that, in the first part of 

 my paper, I shall consider the flight of a golf-ball in a 

 dead calm only, and when it has been driven fair and true 

 without any spin. In a former article (Nature, Sept. 22, 

 1887) I have discussed the effects and the causes of 

 spin. Also I shall confine myself to the " carry," as the 

 subsequent motion depends so much upon purely acci- 

 dental circumstances. By far the most valuable data 

 connected with the subject are those which can be 

 obtained in calm weather alone, and which bear on the 

 form, dimensions, and duration of the first part of the 

 course of the ball. It is mainly due to the excessive 

 rarity of perfectly calm days that our knowledge of the 

 data is so slight. 



Under these restrictions, it is somewhat curious to find 

 that the extreme carry of a golf-ball is not very different 

 from that of a cricket-ball. Both may be spoken of as 

 somewhere about 200 yards. But the circumstances of 

 propulsion are in general very different : — for, unless it is 

 specially teed on a slope, or driven with a spoon (in 

 which case its initial speed is necessarily reduced), a 

 golf-ball goes off at a very moderate inclination to the 

 horizon : — while the sensational drives at cricket usually 

 have the unquestionable advantage of a much higher 

 trajectory. 



Theoretically, the proper position of an observer who 

 wishes to secure at once all the required data should be 

 some miles to one side of the plane of flight, so that he 

 should see the trajectory, as it were, orthogonally pro- 

 jected on a dark background of cloud. The small size 

 of the ball, even if there were not other insuperable 

 difficulties, makes observations in this way impossible. 

 Hence each distinctive feature of the trajectory must be 

 separately studied ; and this implies either a staff of 

 observers, or, what is much less easy to obtain, a player 



NO. 1087, VOL. 42] 



who can make a number of successive drives almost 

 exactly "similar and equal" to one another. I am 

 convinced that many of the great incongruities which 

 I have found among the data furnished to me, even by 

 skilled observers, are due mainly to the fact that the 

 measures of different characteristic features had been 

 made on drives essentially different in character from 

 one another. 



Another fertile source of error lies in the too common 

 assumption that, because a gentle breeze only is felt by 

 the players, who may possibly be in the lee of a sand-hill, 

 there is nothing beyond a similar breeze at heights of 60 

 to 100 feet ; whereas, at that elevation there may be a 

 pretty strong wind. Unless attention is most carefully 

 paid to this, the estimates of the position of the highest 

 point of the trajectory are sure to be erroneous. 



The desiderata which are of real importance ; and 

 which must, if possible, be obtained from one and the 

 same drive : — the air being practically motionless : — are 



1. The initial inclination to the horizon. 



2. The range (on a horizontal plane) of the carry. 



3. The maximum height attained. 



4. The horizontal distance of this maximum from the 

 tee, expressed (say) as a fraction of the range (2). 



5. The time of flight. 



To these we may add, though it is of less importance, 

 and also much more difficult to estimate with any approach 

 to accuracy, 



6. The final inclination to the horizon. 



These data are not independent : — in fact theory (such 

 as it is) shows several • relations among them. But, as 

 no one of them, except the second and fifth, admits of 

 accurate determination, it is desirable to measure as many 

 of them as possible ; so that they may act as checks on 

 one another. 



We may also add, what I have recently been endeavour- 

 ing to obtain : — 



7. The horizontal distance passed over in the first 

 second. 



This, if properly ascertained, would be one of the most 

 directly useful of the whole set of attainable data. 



My experience has been that observers always over- 

 estimate the values of the quantities i, and 6, above : — 

 though they state their ratio fairly well as about I : 3. 

 The time of flight, 5, also is usually given too great. 

 But the greatest over-estimate occurs in the case of 

 datum 4. This exaggeration puzzled me very much at 

 the outset of my inquiry. It is easy to see that, in order 

 to produce a path such as that sketched below, in which, 

 (according to estimates sent me from St. Andrews a 

 couple of months ago, when I was unable to procure 

 them myself) 



AC : AB : : 3 : 4 ; 



(where D is the highest point of the trajectory) the 

 initial speed and the resistance must both be very great. 

 For clearness, the vertical scale is much exaggerated. 



Thus I was led to make some experiments with the 

 view of finding an approximation to the utmost admissible 

 initial speed. This I tried to obtain by measuring the speed 

 of the club at impact, and multiplying by i'6. A hollow 

 india-rubber shell, of the size of a golf-ball, was teed in 

 front of a horizontal axle on which were fixed, six inches 

 apart, two large pasteboard disks with broad borders of 

 very thin white calico. The ball was teed on a level 

 with the axle, midway between the planes of the disks, 

 and three inches beyond their extreme edges. A stout 

 wire, dipped in black paint, projected from the nose of 



