April 5, 1913 
FOREST AND STREAM 
437 
Forest and Stream is the official organ of the National Archery Association. 
Figures, Si^ns and Equations. 
BY E. J. RENDTORFF. 
Confession is good for the soul. Allow 
me, therefore, to admit that until the other day 
I was imalile to spell correctly the word “toxo- 
philite.” As my Webster's dictionary does not 
give the word, 1 hope this fault will be forgiven, 
and that on judgment day the error will not 
weigh too heavily against me. I always was a 
poor speller. 
Allow me in this contribution to apologize 
to the archery world for having inflicted on it 
a series of parodies “fit for comic opera.” 
My effusions have been too numerous for 
modesty. When I get started, in my enthu¬ 
siasm for archery, I seldom know when to stop. 
My articles are an example of this. The case 
reminds me of farmer Jones’ white pig named 
Ink. 
Smith: “Why do you call him ‘Ink’?” 
Jones: “Because he always runs from the 
pen.” 
It would almost seem that I am trying to 
throw a brick at Brother Hertig, whom I fre¬ 
quently quote in this article. This is not the 
case. I simply consider his point of view inimi¬ 
cal to the highest development of the sport, and 
wish to point out to him, and the archery world 
in general, the errors of a mistaken attitude to¬ 
ward the use of simple mathematical and graphic 
analyses of archery phenomena. 
Dr. Hertig takes exception to the equation, 
force X time feet 
velocity =- = - 
mass inches 
He writes: “Get a yew bow and you will 
see, your equation to the contrary notwithstand¬ 
ing, why a liow that gets rid of its arrow most 
quickly is the bow that shoots furthest and 
strongest. The longer the time the string acts 
on the arrow during the twenty-two inches it 
travels in getting rid of its arrow, the slower 
will be the velocit}’ of that arow. As I under¬ 
stand your equation, time entered as an advan¬ 
tageous element in the case of the heavy arrow 
vs. the light one.” 
To a certain extent Dr. Hertig is correct. 
The bow that gets rid of its arrow most rapidly 
is of course the one that imparts the greatest 
, distance 
velocity to the arrow, as velocity =-; 
time 
but in spite of that, time enters as an advan¬ 
tageous element in the case of heavy arrow vs. 
light one. This may seem like another paradox, 
which after all are rather common in this world. 
Thus, the other day I saw two negroes waiting 
for a car. The younger was the son of the 
older one, but the older was not the father of 
the younger. It seemed strange, but neverthe¬ 
less it was true. It all depends upon our point 
of view. 
Let us again consider the equation; velocity 
(of arrow) = 
force (of bow) X time (during which the force acts) 
mass (of arrow'). 
Let us assume that a certain string acts on 
an arrow for iToo second, and through a dis¬ 
tance of l)ut 10 inches; while another string, 
offering the same force, acts on another arrow 
of the same weight for 1/50 second through a 
distance of 20 inches. The latter arrow was the 
force of the string acting for twice the length 
of time as in the first case, and gives a greater 
velocity. 
It is an analogous case to a locomotive start¬ 
ing a train. After the locomotive has pulled for 
10 seconds, a certain velocity is attained; but in 
20 seconds the velocity will be twice as large, 
provided the force of the pull remains constant. 
Let us follow the analogy a little further. 
.After the train has reached its greatest velocity, 
the cars follow the locomotive almost as fast as 
it can move. The force the locomotive exerts 
on the train is then greatly diminished. Simi¬ 
larly, if a man pushes against a sled for one 
second, it will move with a certain velocity, 
which becomes approximately twice as great in 
two seconds. After a while, however, the sled 
moves almost as fast as the man can run, and 
he therefore pushes against it with little force 
and gives it no e.xtra velocity. E.xactly the same 
thing happens to an arrow. The string moves 
forward with a certain velocity depending upon 
the length, strength and elasticity of the bow. 
When the arrow is very light it soon gains a 
velocity almost equal to that which the string 
has when the bow is shot without an arrow. 
I'he force exerted against the arrow is thus 
quite small for approximately the last half of 
the distance it traverses before leaving the string. 
The arrow will then have but little energy stored 
up in it. It will have a high velocity due to its 
lack of weight, hut the friction of the air wdll 
soon absorb its kinetic (motion) energy, and it 
will “peter out” before traveling very far. 
The case of the heavy arrow is as follows: 
On account of its greater weight and consequent 
inertia, or opposition to get into motion, the bow 
w'ill give the arrow a slower velocity than the 
light arrow. The force of the bow, exerted 
through the agency of the string, will there¬ 
fore be greater against the arrow during the 
time the arrow and string remain in contact. 
Furthermore, the time of contact is greater than 
for the light arrow', so that the heavy arrow ab¬ 
sorbs more energy, and is better able to over¬ 
come the friction of the air, or the resistance of 
any extraneous effects, such as those produced 
by variable winds, buckling of the shaft, etc. 
In our equation the product of force X time 
is greater for the heavy than for the light arrow. 
What does this mean ? Why, that the heavy 
arrow will have a relatively greater velocity than 
the light one. Relative to what? Relative to 
its increase in weight. In plain English it would 
mean that if an arrow weighing 220 grains be 
given a velocity of 200 feet per second by a 
certain liow, an arrow' of twice the weight, or 
200 
440 grains, would not have a velocity of - or 
2 
100 feet per second, but in the neighborhood of 
150 feet instead. As the quantity of motion, or 
momentum, a body possesses = mass X hy ve¬ 
locity, it is at once apparent that the heavy arrow 
will have more energy than the light one, over¬ 
come the resistance of the air easier, have a 
more uniform flight, hit the target harder, re- 
l)ound less readily, etc. Although its initial ve¬ 
locity is less than that of the light arrow, it may 
make a better flight shot than an exceptionally 
light one, on account of its greater stored up 
energy and consequent ability to overcome the 
friction of the air. Note, for example, the re¬ 
marks of Randolph Laughlin in his recent article 
on “Archery—Golf’ (printed in Forest and 
Stream, Feh. i ) pertaining to the flight of light 
Turkish arrows. 
Let us ne.xt consider the case of the yew 
bow, which Dr. Hertig believes in some way con¬ 
tradicts our formula. He claims that a 48- 
pound yew will shoot with as great a velocity 
as a 56-pound lemonwood. Perhaps it wilt. As 
I have never shot a yew Ijow I know nothing 
concerning its merits, save what I have been told 
by other archers. Let us admit, for the sake of 
the argument, that a 48-pound yew will shoot 
as strong as a 56-pound lemonwood. If it does, 
w'hy does it do so? 
I believe it will lie clear by this time that 
the force acting on a moving arrow, w'hile in 
contact w'ith the string, is not the full force of 
the bow, and that the force which is operative 
depends upon the speed with which the string- 
moves forward compared with that of the ar¬ 
row. Yew is apparently a very elastic wood, so 
that the bow has a very sharp cast, i. e., occupies 
but little time in moving from full draw to its 
position of rest. In that case the force acting 
against the arrow, after its immediate release, 
would be larger for the yew than for the lemon¬ 
wood, by, say, 20 per cent. The arrow would 
lie in contact with the string for, say, 10 per 
cent, less time. This would make the product 
of force X time 10 per cent, greater for the 
yew bow, and give the arrow that much extra 
velocity. 
The reason why I used the formula will be¬ 
come apparent after reading the following ex¬ 
planation. I was firmly convinced that archers 
are wrong in using light arrows, and introduced 
the equation to show that heavy arrows have a 
certain compensating advantage. Light arrows 
have, of course, a higher initial velocity and a 
lower trajectory up to a certain limit of dis¬ 
tance. This high velocity is desirable and can 
be secured only b}' the use of a light arrow, or 
by a strong bow when used with a heavy arrow. 
