I,— PHYSIOLOGY. 171 
Uniform Speed is the Optimum. 
The amountof energy required per minute to runor to swim, or, indeed, 
_ to propel oneself in any way, increases more rapidly than the speed—in 
_ the cases which have been investigated, approximately as the square of 
_ the speed. This mathematical relation is not exact: the facts can only 
really be described by a curve such as that of fig. 2, but it simplifies the 
argument. From the form of the curve of fig. 2, or from the variation 
of energy output as the square of the speed, we can immediately deduce 
that the most efficient way in which to run a race is that of a uniform 
_ speed throughout. Imagine that a man runs a mile race in 4 minutes 
_ 30seconds at a uniform speed of 6.52 yards per second : his energy expen- 
diture is proportional to 44 times 6.52 squared ; that is, 191.3 expressed 
in some arbitrary units. Imagine now that he runs it at two speeds, 6 and 
7 yards per second, 780 yards at the lower, 980 at the upper speed: the 
total time is the same; the energy expended, however, is slightly greater, 
192.3 instead of 191.3. This small variation of speed in the race has pro- 
duced no serious increase in the energy expenditure. Let us imagine, 
however, that one portion of the race, 665 yards, is run at 5 yards per 
_ second, while another portion, 1,096 yards, is run at 8 yards per second. 
The total time occupied in the race is still 4 minutes 30 seconds. The 
energy expended, however, is greater, namely, 201.5 units. Even this, 
however, is not a very large increase ; by running about half the time 
_ at 8 yards and half the time at 5 yards per second, the energy expended 
_ has been increased only about 5 per cent. as compared with that required 
for running at a uniform speed of 6.5 yards per second throughout. 
Although, therefore, theoretically speaking, the optimum fashion in which 
to run a race is that of uniform velocity throughout, comparatively large 
_ variations on either side of this velocity do not appreciably increase the 
amount of energy expended. 
Possible Advantages of a Fast Start. 
There may, indeed, be advantages in starting rather faster than the 
average speed which it is intended to maintain. The sooner the 
respiration and circulation are driven up to their maximum values, the 
greater will be the amount of oxygen taken in by the body, the greater 
the amount expendible during the race. It is a common practice in mile 
races fo start very fast and to settle down later to the uniform speed : this 
may have a physiological basis in the quickening up of circulation and 
respiration achieved thereby. 
The Simple Mechanics of High-Jumping. 
One final point may be worthy of mention—this time connected with 
_ high-jumping and long-jumping. Recently I made a series of observations, 
_with a stop-watch reading to 0.02 seconds, of the times occupied by a 
number of high-jumpers from the moment they left the ground to the 
moment they reached the ground again. With men jumping about 
five feet the time averaged about 0.80 second. Calculating from the 
formula 
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