I have been thinking about this little question for quite some time now, but haven't really come to a conclusion. It feels a little awkward trying to explain this on a physics forum, but hopefully my ideas will be a bit more understandable on a game forum.

If I were to implement a hit-point system for a game where you lose health based on how hard you get smashed around by different objects, which formula makes a more accurate depiction of the amount of "damage" you receive?

Damage = Change in 1/2 * mass * velocity^2
(formula for change in kinetic energy, 1/2 constant optional)

- or -

Damage = Change in mass * velocity
(formula for change in momentum)

What you probably want is to define "damage" a bit more concretely. An impact hitting a large area (say, a car hitting you) would likely cause far more blunt trauma than one hitting a very small area (a bullet), which will more likely pierce right through flesh and locally shatter bones if it hits them. In fact, the key to mitigating gunshots with bulletproof vests is to spread their impact over a greater area, so i reckon that what you might actually need is pressure, or rather, momentum per area.

Momentum per area to determine whether it's a crushing (large area) or piercing impact (small area), and then just flat momentum to define how powerful the impact is. Kinetic energy rarely makes itself relevant in collisions, actually, if my few physics classes are anything to go by (i stand ready to be corrected by actual physicists). Oh yeah, you might also want a material toughness value where, if exceeded, the hit is a piercing hit, otherwise it's blunt. And then some shenanigans to figure out how deep the hit goes.

Maybe define damage as quantity of material displaced with distinction between stressing and fracturing?

You could define damage as any change to the shape or volume of an object.

However by this definition, moving a limb is technically taking damage.

"damage" is an abstract term, and doesn't make any sense in context.
You mean damage to humans? Look up neurology. It covers any other vague questions you could answer.
If not, you're just trying to split hairs in regular physics.
Things with kinetic energy hit other things with kinetic energy and there is a reaction

Let me revise the question as this:
There is a brick wall and a metal cylinder/rod/object.
The metal object is thrown at the brick wall at high speeds and it makes an impression on the wall.

Disregarding any real life imperfections/imbalances due to shape of object, which formula is more representative of how deep the impression will be?

Depth = Change in 1/2 * mass * velocity^2
- or -
Depth = Change in mass * velocity

Also, this is of course assuming the velocity is sufficient enough to overcome any "threshold" of momentum/kinetic energy required to actually dent the wall.

Depends on the composition and strength of both objects.

That's not "damage" though

Damage is a made up word. You mean impact

Surface area of contact area, some more information about contact, and the duration of the imparted force [how does the force transfer, etc etc, a hammer strike's different from a bullet from an arrow], etc etc.

You'd probably be better off looking at the net energy loss.
For example the energy loss of chemical energy due to an explosion
The bonds that make up whatever object you're looking at release energy when severed. So that should probably be the focus of the formula, I lack the required physics knowledge to help in writing it though.

Also:

Wut.

It's a relative term that denotes a change in something that often illicit pain, or decreased utility [like structure, function, whatever].
Weird thing, anyways.

You can have spherical blasts, which are inverse square functions, or radial blasts which are inverse linear functions, in terms of explosions.

Its an abstract thought I guess. Eh one day words will be eradicated and we'll all communicate through Navajo sign language
im hich as fux

Or with pure aspects.

Sorry only vampires can use auspex. Unless you have the "Werewolf: The Forsaken" handbook, or any other limited edition game scenarios

asdf

I need to savagely delete Opera from my phone and get a better browser