Velocity and energy...
They are related.
When it comes to handguns...
The mass of the projectile is important... and penetration.
Also momentum as mentioned is probably better as well...
Velocity and Mass
Momentum is--> p=mv
So a 9mm round that is 115gr and is traveling 1200fps... has a momentum of...
115gr*1200fps=138000grf/s (grain feet per second) (or 19.714 lbf/s)
a 230gr 45acp at 900fps = 207000grf/s (29.571lbf/s)
180gr 40sw at 1010ftps = 181800grf/s (25.971lbf/s)
Muzzle energy is really Kinetic energy... and is a different formula
Kinetic energy is--> E (sub k)=1/2mv^2 (.5 * m * v squared) (sub k... when written is like the opposite of the exponent in x^2 or x squared, so the k sits below the line rather than above... you could leave it out and just use "E" but it is improper)
A second calculation is needed to properly express the value in imperial/english units. And to turn the grains into the familiar pounds we see in the stats we are given... ft/lb (divide the amount from the above formula by the product of 32.163*7000... 7000gr per pound and gravitational constant of earth in fps... as pounds are not a measure of mass) This extra calculation is why the metric system is superior for this sort of thing... calculations of forces, energy, work... much easier.
(Also... some use 32.174 for the gravitational constant, it changes the numbers slightly, but not enough to really matter. 32.163 is the average local gravitational constant of the US and other areas similarly situated away from the equator. 32.174 is the standard gravitational constant global average and the one used for most computations outside of bullets. US manufactures tend to use the first. Use which you prefer.)
45acp example...
.5*230gr*900fps^2=93150000
divide above by 32.163*7000 (225141) = 413.74ftlb
Differences between momentum and Kinetic energy.
Momentum is directly proportional to velocity and mass. Double either velocity or the mass, and you double the momentum. Momentum also has a vector/direction.
Kinetic energy is directly proportional (or linear) to mass... double the mass means double the energy... but it is exponential to the velocity... double the velocity means 4 times the energy. Kinetic energy does not have a vector/direction.
A 1000lb car moving 1fps and a ball that is 1lb moving 1000fps both have the same momentum... but the ball has much much more kinetic energy
1000lbf/s momentum for both.
15.55ft/lb energy for the car, and 15547.26ft/lb for the ball
So how does all this fit together?
Momentum can be equated to how much time a given force takes to stop the object.
Kinetic energy can be equated to what distance a given force needs to stop the object.
An example using a car... braking distance is related to kinetic energy, and how long in seconds from first applying the brakes till complete stop is momentum.
Momentum is also related to inertia... which is an object's resistance to changes in its momentum or motion. Kinetic energy has no such direct association. Sure as momentum increases, so to does energy, but energy does not factor into inertia in that way.
Mass is also directly related to inertia... mass has inherent inertia, so more mass means more inertia... Mass and momentum team up when the force applied is opposite to direction of motion.
Energy must be conserved within a system...
Meaning momentum and kinetic energy both transfer to the target when hit. When its all said and done, the amount of energy and momentum of all the components involved combined must equal the input amounts.
What matters with bullets is the percentage of each transferred.
What about different bullets?
Rifle rounds and FMJ handgun rounds tend to pass straight through and transfer a small proportion of there energy and momentum.
Hollow points are designed to stop within the target or at least expend more energy before passing through.
For FMJ bullets, both rifle and handgun, the force trying to stop the bullet can be said to be constant. For hollow points, that force increases over time as they expand, up to the point of full expansion.
Energy/momentum transfer creates cavitation of the fluids (and by extension the tissue it is contained within) of the person/animal hit.
Now... since rifles have inherently more energy and momentum than handguns, if a rifle round only transfers 30% of its energy, that is a far greater amount than say 30% from a 9mm... Which is why rifle rounds create more cavitation in the body.
When cavitation exceeds the elasticity of the tissues of the body it passes through, it causes permanent damage beyond the path and width of the bullet... This is why rifles are more effective than handguns. Even hollow point rounds in pistol calibers create relatively little cavitation.
So any bullet that passes through a body must create enough cavitation as it passes through to be effective. (barring direct hits to vital structures)
More cavitation means more damage over a wider area.
Cavitation can be increased by adding drag, which is what hollow points do. Even flat nosed bullets increase drag.
So if we do not have sufficient cavitation to cause a lot of damage outside of the bullet path, we must rely on hitting vital structures directly. This requires accuracy, and the ability to reach those structures via penetration.
Hollow points limit penetration in relation to FMJ.
Well what about momentum vs energy?
Lets move to handgun rounds in effective hollow point designs, in relation to self defense.
We know that the cavitation caused will not be very significant.
They should stop within the body of a person baring any malfunctions of the ammo. So they will dump 100% of their energy and momentum into the target. (or they will at least expend almost all energy before passing through)
Now... rounds with higher muzzle energy will be felt more by the target... I.E. it will hurt more, and the wound path will be larger due to a bit more cavitation.
In hollow point rounds, this more effective transfer of energy and increased pain, helps stop threats. Even if the wound is not fatal, the added pain adds to the psychological aspect/desire to stop when one knows they have been shot.
This effect can not be counted on solely, as some aggressors may be in a state of mind that limits perceived pain and desire to stop when shot.
Rounds with higher momentum will penetrate more, due to their inertia... I.E. Their resistance to slowing down.
Heavier bullets also have higher inertia, so they too take longer to slow down.
And since momentum has a vector (the bullets are moving) more time means more penetration. Penetration is needed to hit vital structures at other than optimum angles.
Adding gains in velocity, which is more effective at rising energy than momentum, can be difficult due to maximum chamber pressures, while adding bullet weight is relatively easy. Sure, you loose some velocity with the heavier bullet, but not enough to matter most times.
Take 2 loads of 45acp... lets say one is 180gr at 1000fps and the other is 230gr at 900fps.
The momentum of the 180gr is 25.714lbf/s
The momentum of the 230gr is 29.571lbf/s
More momentum, even though we lost 100fps and gained only 50gr... to get the same momentum from the 180gr you need 1150fps
This is why "heavy for caliber" tend to penetrate deeper (all other factors being equal, like bullet type/design) This is also why they perform more consistently during tests, and are preferred by many shooters for defense.
The added momentum over lighter bullets, plus the added inherent inertia of more mass, means more penetration. As I said, they team up and work together.
Sure, more muzzle energy also increases penetration, but without the aid of the mass inertia, it tends to not perform as consistently.
This is how I understand the physics/math... I could be wrong... it has been 10 years since college, and I am an electronics tech, so this type of figuring doesn't come up in my day to day life/work. I got bored with nothing to do today, so we end up with this long post. 