Given two arrows of equal momentum, but with one deriving a greater portion of its momentum from mass than the other, the heavier arrow will change velocity (decelerate) at a slower rate as it passes through the tissues. In other words, the heavier arrow will retain a higher percentage of its impact velocity at any given time period during its passage through the animal’s tissues, thus it also retains a higher momentum at any given point during the time required for the arrow to penetrate.
Another way of saying this would be that, though the heavier arrow is traveling slower, it takes a longer time to stop. The result is that the heavier arrow will have a greater impulse of force than does the light arrow.
It is momentum that gives an object in motion the tendency to STAY in motion. The greater the contribution of the object’s mass is to the resultant momentum the harder it will be to stop the forward progression of a moving object. Anyone who has pushed a car in neutral and then tried to stop it will understand this. The more of a moving object’s momentum that is derived from its mass, the more TIME it takes to stop it with any given resistance force.
It is common for proponents of light and fast arrows to counter that the faster arrow will have traveled a greater distance through the tissues in the same time period than will the heavier, and slower, arrow. This would be valid were it not for the nature of resistance forces.
As the arrow’s velocity is increased the resistance does not increase equivalently. The resistance increases exponentially. The resistance of a medium to penetration is reliant on the square of the object’s velocity (assuming objects of a given coefficient of drag; i.e., using arrows with the same external profile, material and finish). In other words, if the arrow’s impact velocity doubles, the resistance increases by a factor of four. If the impact velocity quadruples, the resistance to penetration increases 16 times!
The effect of exponentially increasing resistance is easy to experience. Try holding a hand out the window of the car, while the car is going at a velocity of 30 miles per hour (which is only 44 feet per second), and feel the air’s resistance against your hand. The resistance is very slight. Now accelerate to 60 miles per hour (a mere 88 feet per second). The velocity has only gone up by a factor of two, but the air’s resistance to your hand passing through it is now four times greater.
Now imagine the effect on an arrow passing through tissues. Tissues are more solid than air. They have a greater density. Their resistance to an object’s passage is higher. Visualize the effect as an arrow’s velocity increases from 150 feet per second (a fairly typical velocity from a mid-draw weight traditional bow) to 300 feet per second (as from a top line compound bow).
Let us now assume an arrow weighing 700 grains for the slower bow (150 fps is easily achievable with that weight arrow and a ‘traditional’ bow) and a 390 grain arrow for the faster bow (the advertised velocity rating for one of the newest compound bows on the market, using that weight arrow). The slower arrow has 0.466 slug feet per second of disposable net force. The faster arrow has 0.519 slug feet per second.
Lets also assume these two arrows are of same materials, have equal physical external dimensions (easily achievable), and both have perfect flight characteristics. The tissue’s resistance increase is totally dependant upon the velocity of the arrow.
The lighter arrow has 10.22 percent more disposable net force (and 123.2 percent more kinetic energy) than the heavier arrow but, because of its higher velocity, it is met by four times the resistance to penetration. Which arrow will penetrate further in real tissues? Empirical evidence from the outcome studies provides an overwhelmingly definitive answer. Both the frequency and degree to which the heavier, slower, arrow out-penetrates the lighter one is of such a magnitude that it must be viewed as the norm.
