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I'm writing a story about mermaids. For this reason, I'm wondering how much you can hurt someone by say, throwing a punch, hitting them with a rock, or even throwing a spear or rock.
I'm not sure how to calculate it. I'll need to work out the drag forces that hinder acceleration, as well as quickly reducing the speed of the attacking object.
Can someone help me crunch the numbers for throwing a punch underwater, which can be used as an example for my other calculations?
Melee type combat has relied on a variety of mechanisms of injury over the millennia. We have lots of options because Air does not provide significant resistance to rotational motion. This makes all sorts of things like clubs and maces practical, slashing weapons like swords, Trebuchets and Catapults rely on it too. All of these get pretty well taken off of the table due to the resistance to be found in water. Not so much from the weapon itself, but from the arm or armature.
That leaves you with stabby weapons. Things that produce puncture wounds would be best. This is because the cross section of such weapons would not have any appreciable fluid resistance and they are still effective with slower body motions. Because of this, Your Merfolk are going to evolve tactics based around this. The math is not really necessary.
Other answers have brought up various projectile weapon options, and they did it better than I could. Elastic band or spring driven would be best. Just keep in mind that what we think of long ranged and underwater long ranged is going to be different. A system that could launch a Javelin a couple hundred meters in air may only have 20 meters effective range in water. That is going to have an impact as well.
Firearm kinds of weapons are going to have major issues. Rifle rounds fired into water slow down to harmlessness in a few yards. The same rounds are often used to take Deer at a few hundred yards.
Anything you need to go a long way is going to have to have to have some sort of mechanism to continuously propel it forward. Like torpedoes. If you think about it, torpedoes are the only long range weapons that travel underwater to their targets.
Explosives as area effect weapons are still viable. Even rednecks know this. Dynamite fishing anyone? Explosives cause pressure waves that can kill. There are a number of mythbusters episodes that show how this works. Shrapnel is less of a problem underwater.
You have options, but they are limited. On the other hand, remember that Underwater you have a 3 dimensional field. Most human battlefields are fought on the surface of the planet. There is a limited up or down component. Even a lot of air combat is based around hitting targets on the ground. Fighter dogfights happen because fighters are protecting the ability to drop bombs on the ground.
Have fun with the fishes, just don't sleep with them.
Having anyone to fight underwater with the same strategy as we know it would be foolish.
You can have javelins travel quite fast in water, but you need to devise some other way (e.g.: rubber bands) to throw them because using your arm or a bow will encounter too much water resistance before you manage to throw it.
Much better would be to use a lance (i.e.: to swim toward your foe with weapon "on rest", so that your body inertia give it power).
In general you need to rely on mass and its inertia to harm, not speed.
Any projectile should be "harpoon-like", heavy with small cross-section.
Any throwing device should not have swinging parts (use elastic cord, spring or compressed air).
It is not by chance that spearguns are made that way.
I mean easier than doing the calculations, you can go underwater and try to throw a punch, and you'll see it doesn't really work. You generally need a weapon to fight underwater, and then a cut basically kills as it is really hard to stop bleeding underwater.
