The impact of a punch

Main Question or Discussion Point

I have searched online and found that the power of a punch P=Fv, where F is the force of the punch and v is the speed. But in my mind it works like this: let's just say that only the arm speed is taken into account, then before impact, the energy of the arm travelling is E=(1/2)mv^2. That is how much work the target has to do to stop the punch. Therefore, the impact force Fi = E/L where L is the displacement before the arm reaches zero speed.

In conclusion, the impact of a punch is proportional to m & v^2?

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Yes, in short you are correct that the impact force of a punch is proportional to the mv^2 of the arm. Of course, there are some additional small corrections that you could apply due to deformation effects. For example, the stopping displacement L is not a constant, but will increase slightly as the energy increases, because a good solid punch deforms more flesh than a soft one. Also, the arm and fist are not nondeformable objects, so some energy will also be lost in them as well. Hitting a brick wall will put more energy into your hand than it will into the wall.

Back when I was taking Karate, our sensei would often try to exhort us to punch harder and faster by quoting things like "Power = Force times velocity!". Of course, in physics words like power and force have specific meanings, which aren't always the same as they mean outside of physics. When punching, force or the energy absorbed by the target are more important than power. Consider these 3 cases, all using the same amount of power a) a fan blowing air (small force, high velocity, high deformability), b) a guy punching you (medium force, medium velocity, medium deformability), c) a guy punching you wearing brass knuckles (medium force, medium velocity, low deformability). Which you would rather be hit with?

Thank you!

The low deformability of the "weapon" would transfer more energy to the target, and less to itself. It would make sense that the impact would be stronger with less deformable weapons. Also, the less the contact area the more the impact.

The force of a Punch:
It’s easy to start from the equation that force equals energy divided by distance to confirm that the force is proportional to the kinetic energy of the punching fist. Since kinetic energy is (1/2) mv^2 you can then see the importance of the velocity of the strike and why all martial artists want to put body weight behind their punch (or why a good kick can finish a fight because it’s both fast and ‘heavy’). The distance referenced in F = E/d is the distance travelled during the impact to expend the energy referenced – e.g. the penetration into the target. But that’s not necessarily the end of the story. What you may be interested in is the deformation energy that’s imparted into the object that’s struck – useful when calculating if you can break a certain size and type of wooden board. To derive that relationship use the conservation of energy and conservation of momentum equations for the collision of two bodies – such as a fist and board. Then if you really want to be precise include a coefficient of restitution that reflects the hardness of the bodies being struck. Best reference for this type of physics of kicking and punching is the book “Parting the Clouds” by Grenville Harrop.