I'm doing it as hard as I can.

[Lorizael]

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.

[ZEE]

That's what she said.

[Lorizael]

If you can find me a woman just as fascinated by this puzzle as I am (and also hot, single, roughly my age, child-free, and for some ridiculous reason willing to talk to me), I will donate $100 to a demonic charity of your choice.

[Garth Vader]

I assume the statement "as hard as you can" refers to as fast as you can.

As for how hard something is going to hurt for a given mass, there is going to be a sweet spot where it's possible to throw to a decent velocity and heavy enough to hurt and it's going to be somewhere heavier than a ping pong ball, but lighter than a big bowling ball.

[Dinner]

With as much force as he can?

[rah]

Yeah, I agree that there will be a range in terms of weight where you can maximize the velocity. I don't know how narrow that range will be, but I suspect it may be larger than you'd think.

[Elok]

I think you're incorrectly assuming that the effort scales in a linear way, where twice the effort produces twice the outcome. It's more likely that, for an object twice as dense, your muscles have to work much harder for diminishing returns. Note also that there are a LOT of muscles involved in all but the most trivial actions, and any one of them could be a limiting factor.

[Lorizael]

I assume the statement "as hard as you can" refers to as fast as you can.

Doing some googling, the biomechanics of a baseball pitch talk a lot about the angular velocity of various body parts during the pitching action, so that definitely might be part of it. In that case something like torque is probably the relevant factor.

[Lorizael]

I think you're incorrectly assuming that the effort scales in a linear way, where twice the effort produces twice the outcome. It's more likely that, for an object twice as dense, your muscles have to work much harder for diminishing returns. Note also that there are a LOT of muscles involved in all but the most trivial actions, and any one of them could be a limiting factor.

Yeah, my hope was that I could get away with brushing all that under the rug spherical cow style and still get a reasonable answer. Maybe not.

[Elok]

I'm not sure it works even for mechanical engines. Do you use 20% more fuel maintaining a speed of 60 mph in a car as opposed to 50? How do the costs of acceleration work out? I don't think it's all that elegantly Newtonian.

[Lorizael]

Oh, it's very Newtonian, it's just not linear. For example, at higher speeds, drag starts to become an issue, with the force from drag being proportional to the square of the velocity.

[BeBMan]

For bringing KH back we need to add some Jennifer Lawrence refs to our posts.

[Lorizael]

I'm doing it as hard as I can.

[BeBMan]

I wish there was also a way to bring Loin back. No joke.

(edit: ok, maybe I shouldn't derail this thread)