For a fixed power input, is it better to hit something with a light rod or a heavy rod?

Question

My friends and I got into an argument about whether it was more damaging to hit an object with a heavy rod or a light rod. The sides of the argument go something like:

"If you swing the light rod, you'll be able to swing it much faster and therefore do the most damage!"

"Yes, but if it's too light there won't be enough mass behind it and it won't damage as much!"

I have been toying around with this question for a few months and I have yet to come up with a satisfactory answer. I'm setting up the problem like this:

Person A is swinging a rigid rod of mass $m$ and fixed length $L$ at Object B located at a fixed distance $d$ away. Imagine Person A is swinging the rod much like a baseball player would swing to hit a ball. Person A has a fixed power $P$ to input to the rod over distance $d$ before making impact with Object B.

Given input power $P$, rod length $L$, and swing distance $d$, what is the optimal mass $m$ of the rod to maximize the damage of Object B?

For simplicity, I'm modeling the power application to be over a linear rather than angular distance, and I'm modeling Object B as a spring with spring constant $k$ attached to a fixed wall. Effects of gravity should be ignored. Also, for now ignore the constraints of human strength and assume we can swing anything as heavy as we like.

Answer 1

It's better to be hit more slowly with a heavier object.

Your comment to Raindrop's answer pinpoints the issue. When the object hits you, your flesh reacts to the impact with a certain timescale. With slower velocities your flesh moves out the way, but at higher velocities the flesh cannot move fast enough and you get localised damage. To illustrate this imagine being hit slowly with a lead pipe or fast with a whip. The slow impact with the lead pipe will do little damage but the whip will cause inury. I can speak from experience having been hit by a whip whilst fooling around at a riding school, and it flipping hurt and raised a weal!

However your question is physically a bit unrealistic. I cannot put the same amount of energy into a riding crop as a I can a lead bar because there is a limit to how fast I can move my arm. That means I can do a lot more damage with a lead bar than with a whip simply because it's possible to concentrate far more energy in it.

You also need to be careful to eliminate the effect of gravity. Unless you swing your cudgel exactly horizontally there will be some contribution to its energy from gravity. Because the force exerted by gravity is proportional to the mass of the object, the heavier object will end up with more energy and do more damage.

Answer 2

Think of it this way: we have to lift the rod up first, and then down (so that it's faster) If you have a heavy rod, you get to add lots of energy to it when you lift it up (E=mgh) So when you're bringing it down, assuming the time for impact, $t_{impact}$ with the person you're beating is same for both light and heavy rod, you want to maximize the force exerted on that person, $F_{impact}$.

Note the equation for change of momentum, $$\Delta p=m\Delta v=F_{impact}t_{impact} $$
if $m$ is larger, then $F_{impact}$ is larger

If you swing it at constant height, $W=\int P dt$ so the more time you have to swing the bat, the more energy you put into the bat. Since it would take a longer time to swing the heavier bat (F=ma, it accelerates slower), you get to put much more energy into the bat before it hits your target. So it'll have more kinetic energy $0.5 m v^2$ which means larger $mv$ which means larger $F_{impact}t_{impact}$ which means larger $F_{impact}$.

An 'optimal mass' would depend on gravitational force $GMm/r^2$. You want the mass to be such that it is as large as possible without missing your target (e.g. falling to the ground before hitting your target) This is a trajectory problem, if your power is applied horizontally (and doesn't help your bat to move upwards) then your optimal mass will depend on the power supplied, $P$.