So I've been contemplating something of a biomechanical riddle and I want to see what others think.

What does it mean to throw something "as hard as you can"? That is, what quantity is being maximized in a throw that is as effortful as you can manage? The two that spring to mind are force and energy. But neither of these seems satisfactory because of what my intuition tells me about this situation. Consider what happens if you throw ball A as hard as you can and also throw ball B, which is twice as massive (but aerodynamically equivalent), as hard as you can. What difference does the mass make?

My intuition says that if I hit you with ball B, it's going to hurt more.

My intuition also says that if I throw ball A as hard as I can 10 times, and throw ball B as hard as I can 10 times (the next day...), I will be more tired after the B throws.

Here's where we run into problems. Assume "as hard as you can" maximizes the force. In that case, the momentum imparted is the same for each ball and consequently ball B will be moving half as fast. The kinetic energy in ball B, on the other hand, is half the kinetic energy of ball A, because that's proportional to v^2. So for two balls thrown with the same force, both balls impart the same momentum, but the more massive ball delivers less energy. How then does the more massive ball hurt more upon intact?

Okay, then let's change it up and say "as hard as you can" maximizes kinetic energy imparted to the ball. Energy upon impact is the same, but now the momentum of the heavier ball is greater by a factor of sqrt(2) (the same math but working backward). That's great. The more massive ball imparts more momentum, which makes sense if it's going to hurt more. But... you're expending an equal amount of energy in each throw, which means throwing ball B a whole bunch is not going to tire you out more than throwing ball A a whole bunch. And that doesn't seem right.

So what's the solution here? Either my intuitions are not correct, or some other quantity is being maximized, or something about the situation changes for balls of different mass. Thoughts? Really hoping to get JM, KH, Ramo, and Rogan Josh in on this.

What does it mean to throw something "as hard as you can"? That is, what quantity is being maximized in a throw that is as effortful as you can manage? The two that spring to mind are force and energy. But neither of these seems satisfactory because of what my intuition tells me about this situation. Consider what happens if you throw ball A as hard as you can and also throw ball B, which is twice as massive (but aerodynamically equivalent), as hard as you can. What difference does the mass make?

My intuition says that if I hit you with ball B, it's going to hurt more.

My intuition also says that if I throw ball A as hard as I can 10 times, and throw ball B as hard as I can 10 times (the next day...), I will be more tired after the B throws.

Here's where we run into problems. Assume "as hard as you can" maximizes the force. In that case, the momentum imparted is the same for each ball and consequently ball B will be moving half as fast. The kinetic energy in ball B, on the other hand, is half the kinetic energy of ball A, because that's proportional to v^2. So for two balls thrown with the same force, both balls impart the same momentum, but the more massive ball delivers less energy. How then does the more massive ball hurt more upon intact?

Okay, then let's change it up and say "as hard as you can" maximizes kinetic energy imparted to the ball. Energy upon impact is the same, but now the momentum of the heavier ball is greater by a factor of sqrt(2) (the same math but working backward). That's great. The more massive ball imparts more momentum, which makes sense if it's going to hurt more. But... you're expending an equal amount of energy in each throw, which means throwing ball B a whole bunch is not going to tire you out more than throwing ball A a whole bunch. And that doesn't seem right.

So what's the solution here? Either my intuitions are not correct, or some other quantity is being maximized, or something about the situation changes for balls of different mass. Thoughts? Really hoping to get JM, KH, Ramo, and Rogan Josh in on this.

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