68 



DISCOVERY 



the square of the speed. In the figure are shown the 

 world's records, taken from Whitaker's Alman'ack, of 

 flat races from loo yds. to i mile. Horizontally are 

 plotted the times occupied in the race, vertically the 

 squares of the average speeds at which the record was 

 made. The curve joining the points is a fairly smooth 

 one, though it is noticeable that the principal records 

 (100 yds., 220 yds., J, |, and i mile) lie rather above 

 the curve as a whole. More effort has been expended 



100 



gives approximately the relation between the rate of 

 oxygen consumption [not supply) and the time occupied 

 in the race. Now if the maximum oxygen " income " 

 of a world's-record-holder be 5 litres per minute, and 

 his maximum oxygen " credit " be 12 litres, then in 

 various times he can take exercise equivalent to the 

 following qtiantities of oxygen : 



Time : minutes . . i 



Total oxygen available ; litres 1 7 

 Oxygen per minute : litres . 17 



3 



27 



Q 



4 

 32 



5 

 37 

 7-4 



62 

 6-2 



50 



We see that the oxygen available per minute is much 

 less in a long race than in a short one ; hence the speed 

 must be less too. Indeed, if the oxygen per minute 

 in the above table be plotted against the time, it gives 

 a curve of just the same type as that shown in the 

 figure. This curve can be given a simple mathematical 

 equation. Let x be the time occupied in the race and 

 y the average rate of oxygen consumption per minute. 

 Let A be the maximum oxygen " debt," and B the 

 maximum oxygen "income." Then assuming for 

 simplicity that respiration can start at its highest 



Ll-I 

 00 



oe 



U-l 



D_ 



CO 



en 



<: 



Dotted curve from equation 

 ■y= m + lioojx 



<j> world's records, ioovzzo yos.,1/4,'/z %■ imile 



O ditto , OTHER DISTANCES. 



20 40 60 80 



SECS. 



200 



on these records, and it is harder to beat one of them 

 than one of the others ; indeed, simple inspection of 

 the curve shows that the easiest records now avail- 

 able, for any enterprising and scientific would-be 

 record holder, are those for 150 and 500 yds. What, 

 however, is the cause of the general form of the 

 curve ? It is simple to show that, apart from one 

 factor, the principles we have discussed above (of 

 oxygen "income" and o.xygen "credit") provide a 

 satisfactory general explanation. 



In the first place, the reader will notice that, in the 

 figure, we have plotted the square of the average 

 velocity ; as we have pointed out above, this quantity 

 is proportional approximately to the rate of oxygen 

 consumption per minute, so that in effect the figure 



value immediately " on the pistol," and that the 

 runner runs at a constant speed (both only approxi- 

 mately true), the following relation must hold: 



Total oxygen = yx ^ h. -{- Ba; 

 or 3* = B + Ajx. 



In the figure an equation of this type, represented by 

 the broken line, is shown to fit the curve with reasonable 

 accuracy. It is noticeable, however, that at short 

 times the relation does not hold at all. The reason 

 is obvious. In the shorter races it is impossible to 

 exhaust all the oxygen available, even in the most 

 violent effort ; the limit is imposed here not hy the 

 oxygen supply, hut by the sheer capacity of the muscle. 

 There is one fundamental weakness in the data used 



