This entry will serve as an intro to at least two subsequent articles: “The Physics of Guns: Present and Future” and “Newton’s Folly and
the Power of V Squared,” a discussion of the importance of kinetic energy and
Newton’s failure to understand it, along with historiographical commentary.
Blunt force trauma is measured by kinetic energy, not momentum. An object's ability to transfer force to a target is more influenced by velocity--and, thus, kinetic energy--than mass and momentum.
A simple verbal argument: If an object has more energy, it should tend to TRANSFER more energy; if you put more (work) into it a ball, you should get more out of it--energy transfer. A higher energy transfer means a greater increase in the velocity of the Target and, thus, delivered force.
But more rigor is required.
Imagine a human body as a target (T) at rest with respect to
the ground being struck on two separate occasions by different bullet-shaped
objects. H is twenty pounds and moves at five miles per hour, and L is a
quarter-pound moving at four-hundred miles per hour. Both have
equal density and momentum**; H is heavier (with greater length) but
lower velocity and, therefore, less kinetic energy. But it should be
intuitively obvious that the quarter-pound ball is going to cause more damage.
**It’s imperative to vary dimensions and not density because
varying density makes the notion of mass more ambiguous, which is obviously
problematic in a thought experiment comparing the effects of momentum and
kinetic energy.
Noting that the object is merely SHAPED like a bullet and
will not penetrate the target’s flesh and that both collisions are elastic, I
postulate that the human target (T) will be accelerated more in the T-L
collision (than T will be in T-H) and, therefore, (T, the target itself,
will) be subjected to more momentum.
It should be intuitive, given its far lesser mass, that L
will be decelerated more than H (for any given force, H will be
accelerated/decelerated 1/80 as much as L). The T-L system has more energy,
which is conserved. The only way energy (and momentum) can be conserved is to put more
velocity into T (compared to the amount in the T-H collision), and this
requires more force.
But given the fact that L will slow down more than H and momentum is always conserved, even if the collision were inelastic, T would still be subjected to a greater force.
For a given force, an object’s CHANGE in momentum is based
solely on its mass. If its initial momentum is higher, the change will leave it
with a higher final momentum, but the MAGNITUDE of the change is unaffected, so
long as you’re dealing with the same force. Though the force the moving body DELIVERES
to a second body will be higher when the former is initially moving faster, so
long as you know the force TRANSFERRED TO the body, the initial velocity has
nothing to do with its resistance. But its velocity squared tells you more
about how much force it can deliver.
Proof for where
the bullet gets stuck in the target (and becomes single masses).
Because both systems have the same momentum before and after the collisions, the mass of the L and T system is smaller and, therefore, has to make up for it with more velocity, following combination. U is the velocity of H+L, and V is the velocity of L+T. (H+T)U = (L+T)V. Thus, the ratio of the velocities of the combined objects are inversely proportional to their masses. Yes, the mass of H+T is bigger and, if all factors were held equal, it takes more force to accelerate it the same amount, but the concern was the change of velocity of T, the target. T was subjected to more force in the T and L collision because it accelerated more.
The conservation of momentum, though an indispensable
physical law, is still just a means of finding the initial or final velocities
(or masses) of bodies that are accelerated relative to the ground exclusively
by INTERNAL forces. KE generally tells you more about an object
than its momentum. This argument is reinforced by the fact that if force is
applied over the same distance, it takes four times as much force to double the
velocity of an object (or that amount squared). This also means that, all factors held equal, a bullet moving twice as fast will penetrate four times deeper into a body.
KE's relationship with work is, of course, more important than its ability to deliver force in a collision, but the latter fact gives much further meaning to the notion that KE is not JUST a reflection of the work that HAD BEEN done to it, but also its ability to TRANSFER mechanical force in addition to energy.
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