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Lets say a character can create magic propulsion to push themselves, fly, and move quickly. And then this character uses this speed and momentum to punch someone. At what speed would the user begin to take damage from doing such a punch like breaking their bones.

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Figure out damage at speed by the speed of an equivalent free fall.

5 feet

60 feet

injury score by fall height

You could figure it out. Figure out how fast your character is going when he hits. Then figure out how far one would need to fall to be going that fast using the fall calculator. https://www.omnicalculator.com/physics/free-fall

Then plot what sort of damage is done using the graph which I took from here: Fall from heights: does height really matter?

- I think it is safe to say that going faster than 15 mph will break a bone on impact. Whether it is your hand or your hand and arm, or hand arm and shoulder depends on how fast you are going.

When I am flying in to give someone a whack I am going to be holding a rubber chicken, and holding it loosely. That equal and opposite reaction is no fun at all.

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In a bare knuckle fight, which I assuming is the premise, punching someone in the head is more likely to break your hand than their head. Even with a lot of training applying a closed fist to the bony structures of the face often results in broken and dislocated knuckles and bones in the hand, no extra momentum need be applied. In my personal experience most of the training in martial arts that I have received, whether it be boxing, Ju Jitsu, or Kung Fu focuses on not hitting people in ways that expose the bones of the hands and feet to solid impacts on the bones of your opponent. I would imagine that you risk breaking bones on the softer targets, like the stomach and solar plexus, in much the same way that one will mangle themselves punching someone in the head if the impact speed were increased by two thirds to double, i.e. if the person throwing that "punch" were traveling 25-30mph and plowed an upraised fist into their target.

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If they're unarmed? Zero. As mentioned elsewhere, if you punch someone in the head, you're more likely to break your hand (this is actually what boxing gloves are for - protecting the wearer's hands). If you can fly and want to hurt someone, just drop a rock on them.

If they're armed, though, it's a different story. A blunt weapon probably has the lowest speed limit - above some speed, the impact force will jar it out of your hands. Professional baseball players can hit 150km/h fastballs with no trouble. But baseballs are designed to be easy to hit, and are so light they offer very little inertial resistance - more massive targets, like someone's head, would be much tougher (if you want a ballpark, picture using a baseball bat to hit a baseball, a soccer ball, and a stop sign).

One solution is to make the blunt weapon heavier. This way, you do most of the work by accelerating up to speed - when you hit, you aren't swinging the weapon so much as you are traveling next to a large, heavy object that happens to have a handle you're holding on to. For example, a car hitting a human at highway speeds will have a much greater inertial effect on the human than the human will have on it. If this character can use his magic propulsion to lift very heavy objects, they could just throw a car at their target... but then they wouldn't really need to be anywhere near what they were throwing. So the maximum speed and thus power of a blunt weapon strike will be determined by how much they can lift and carry with them.

But if you really want to hurt someone by moving fast, you want an edge. Literally.

Bladed weapons bypass the impact force limitation, because they don't care about impact force - in fact, less force is usually better. With a reasonably sharp blade, you only need a few pounds of force to cut skin. Contrary to what Errol Flynn's filmography would suggest, most fights involving bladed weapons are over very quickly - even a glancing blow can cut enough to be quickly fatal. If your flying character has a sword - or even a knife - they wouldn't need to be moving very fast to kill or seriously injure their target.

The downside of blades, of course, is that they are much less effective if the target is wearing any kind of armor. Some armor has gaps or limits that a well-aimed sword strike can get around, but if the target is wearing something equivalent to a full suit of plate, then you're going to be out of luck. In this case, your character would be back to blunt weapons. Many works focus on piercing weapons (such as lances or war picks) as a means of defeating armor, but those would be much less useful for your flying character - it would tend to get stuck in the target and yanked out of their hands.

In conclusion: Bypass the problem with swords and/or dropped rocks.

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