Every time I've heard physics invoked to explain how to hit hard I have been dissatisfied. The most common thing is to invoke F=ma --- force equals mass times acceleration, and talk about how if you want some more force, you gotta get some more speed. Or, put some more mass behind it. Force, yeah! Even if they get beyond that, and explain, well, acceleration is not exactly the same thing as speed, and the force here doesn't actually mean anything, and hedge a bit, and then mention you're only describing the thing you're hitting with, not the target, but equal and opposite reaction, right? Force on target!
What you're talking about, when you talk about how you're throwing a punch with F=ma, is how much work YOU are doing. Acceleration is velocity per time --- a description about how long it takes to make something go a certain speed. The mass is how much you're moving. The force is how hard it is to do it. At the end of that process, you've launched a set mass with a certain velocity, and if you want that velocity to be higher, or for that process to take less time, you gotta work harder. It's just about you, it's not about what that hit is going to be like on the other side.
For striking combat sports, that actually does a decent job of describing what you want. You're basically throwing a ball at a bell at a bell. Whether that ball is a fist or the scoring tip of a foil, the goal is the same- you want to do that with minimal telegraphing (highest acceleration), low commitment, easy to get back on guard (just the mass of the ball, no more), with enough velocity to ring the bell (depress the tip, meet whatever is required to score). For this reason, snapping actions are king. You get a kind of wave of motion through your body to launch that ball, and then pull it back after you score. Or let it continue circularly to come back after glancing off.
That's all pretty clear. And you can sort of expand that to see how if you put more mass behind it, you can hit harder. But it doesn't explain the variety of hitting effects I've seen demonstrated, and the damage they can cause. What more do you need?
First, you switch your thinking. that F=ma? You have to think about it as force equals mass times deceleration. How hard the surface has to work to stop that ball. And your thinking about that mass has to change, too. A little example:
Imagine you have a piece of paper falling down as a vertical plane. Shoot a needle at it. The needle punches through. Throw a marble at it. The paper just sort of crumples out of the way. What's going on? Well, part of it is surface area. But. Suppose we attached big weights to the corners of the paper and dropped it. The needle punches through, the marble punches through. We've increased the mass of the paper. Mass is inertia, base resistance to movement. If it's moving, it's not getting damaged. In order to damage something, we need to do that damage before it starts moving away. If something has more mass, there's more potential for it to be damaged. Momentum is inertia with added effect with velocity. If something is moving toward the striking object, there's more potential for it to be damaged. Consider: Would you rather be hit in the face with a baseball traveling 49mph while you are starting to walk forward at 1mph, baseball traveling 50 mph while you are standing still, or 51mph while starting to fall backwards at 1mph?
So then, damage. This is where things get complicated, and you do not see equations invoked casually. If you wanted to learn how to break things, you'd think you'd want to know how things break, but... it's complicated. From my brief survey on the topic, you've got elastic deformation, plastic deformation. Stretching. Fracturing. Structural deformation, structural collapse. Each of these governed by equations that vary by thickness, constants different for different materials. Humans bodies are mix of elastic and inelastic materials, in a combination rigid and flexible structure. It's a mess! Some takeaways:
Elastic deformation is where something bends or compresses, but returns to shape afterwards, also returning the stored force. Not damage. Skin and muscle can deform elastically. For combat sports, this is what you want. No one gets damaged, and also that returned force helps you get back.
Plastic deformation is where something bends or compresses, and does not return to shape. A form of damage. Can't think of anything on the body that does this. Metals do.
Stretching can be elastic or inelastic. A combination of deformation and stretching is what allows for force distribution, like foam. Parts compress and pull other parts in to compress. Time is important- if force is applied quicker than the material can take, the material rips instead of stretching, and force distribution stops. I think this is what's going with the guys who can make you feel knuckles through boxing gloves, despite the fact that there are bigger guys who can hit "harder".
Muscles are somewhat elastic, ligaments and tendons less so. Bones are slightly elastic. "Brittle" is actually a function of stretching and compression- when something bends, it has to get longer on one side and shorter on the other. Once it can't do that, it's going to break. Both length and thickness are a factor.
Some materials and structures, columns especially, can withstand much higher forces briefly and suddenly applied than they can if the weight was slowly applied or left in place. Consider someone crushing an empty aluminum can on their forehead. There is a dead zone between pushing strongly but slightly slowly and quick crushing where you can really hurt yourself.
Torque (turning force) has a tremendous influence on structures. If you didn't crush that can on your forehead (good choice!), you can do the fun thing where you can set it upright on the floor and slowly and carefully stand on it. Then have someone tap the side with a pencil or twist slightly.
Some structures have secondary structural solutions. That is, force is exerted, they're stable, force increases, they crumple more with more force, then hit another stability point where they hold for awhile.
Oh yeah and equal and opposite: anything you hit with is subject to the same deformation and crumpling and whatnot, and all that decreases the damage done.
hmm. to sum up a bit. To evaluate the damage potential of an attack, you have to look at the pressure (force per area) for what amount of time, with what torque, the material of the target for elasticity and force distribution, mass and structure of the target for resistance and crumpling.
We have a lot of experience with not breaking things. We have an unconscious understanding for how to touch and handle and drop a wide variety of things without damaging them or ourselves. Most of us do not have much experience with breaking things. Maija mentioned in the comments on the last post about Muay Thai fighters sparring light and having no problem going for real. True, but the real there is also still a combat sport. Taking a muay thai kick on the thigh really, really sucks. It's disablingly painful and a fight ender, for sure. But that same kick aimed slightly lower with a little torque would destroy the knee. Or kicking the knee itself. And I'm pretty sure with a little training many Muay Thai fighters would be able to break the femur, wherever the kick. but it's not in the sports training, and it's against the rules to hit too hard in a Muay Thai fight (weird, right?). The training spreads the force out just enough to hit as hard as possible without causing damage. There's a big difference between punching the air, and punching a heavy bag. There's another big difference between punching a heavy bag and punching something that will break or deform. Or crumple, grab, and counterattack.
A science fiction book I read had a punching bag with a programmable smart gel that would provide progressive resistance up to a set density, like bone, where it would release. I immediately wanted one, but I think the martial arts world would be a very difference place if that technology was everywhere.