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Another question from my textbook, directly following after this question:

2 identical bullets are fired, on from a lighter rifle and one from a heavier one, with the same amount of force. Which rifle will hurt the shoulder more?

First off, I'm not even sure if this is an appropriate question for Physics.SE as it deals with what 'hurts' our bodies more. But since we're essentially determining whether acceleration or force matter more, I thought it could be asked.

So on with it.

(1) Let the lighter rifle be denoted as $l$ and the heavier one as $h$; [Assignment]
(2) The force exerted by the rifle on the bullets is the same; [From Q]
(3) Every force has an equal and opposite reaction; [Axiom]
(4) The bullets exerts an equal force on the rifles; [From (2), (3)] $$ (5)\ F_l = F_h \ \ [From\ (4)] \\ (6)\ F = M \cdot a \ \ \rightarrow \ \ a = \frac{F}{M} \\ (7)\ M_l < M_h \\ (8)\ a_l > a_h $$

That was easy. Both recoil with the same force, but the lighter one is accelerate faster than the heavier one.

But the point is, does higher acceleration mean more hurt? Or is it force that matters? If the former, the lighter rifle will hurt more. Else if the latter, then both will hurt equally.

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3 Answers 3

Both will exert the same impulse on your body, since this is equal and opposite to the impulse exerted on the bullet, which was stipulated to be identical in both cases.

Impulse is force x time. The difference will be that the lighter gun will push you with a higher force for a shorter time. This will make the impact feel sharper, which can make it hurt more. If you don't believe that, consider this effect taken to the extreme. Let's say something pushes at you with 20 pounds for 100 ms. Let's say that "hurts". What if the same impulse were spread out over time, like 2 pounds for 1 second? If I pushed your shoulder with 2 pounds for 1 second, I doubt you'd say that "hurt". What about 1/2 pound for 4 seconds? Clearly spreading out the impulse over a longer time makes it "hurt" less.

One way to look at the mass of the gun is that it acts as a low pass filter to the very strong but short impulse from pushing the bullet. This causes the strong but short force to be spread out as a weaker but longer force. Momentum is still preserved since the force x time product is the same in all cases.

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I don't think this is a force problem, but a momentum conservation/impulse problem. The first thing I do is notice that the 2 bullets are identical and fired with the same force, thus they have the same velocity after being fired (assuming the time whereby the force is acting on the bullets is the same in both cases). So, by conservation of momentum, we have:$$M_lV_l=mv$$ $$M_hV_h=mv$$ What this means is that the momentum of both heavy and light rifle are the same, and so the same impulse is required to stop them.

So which one hurts the most? It all depends on how much force you use to stop it, or likewise how much time you take, by the impulse equation:$$J=Ft$$

It seems most likely that it will hurt most when the force is high and the time is low, like a punch. On the other hand, a small force extended over a longer period of time will not feel so bad, like a pat on the back.

Does the lighter rifle require more force to stop, as compared to the heavier rifle? I feel that if the shooter can decide with how much force to decelerate the bullet, then both should feel the same. Though if you take it to the limit, a very fast and light rifle would seem to hurt more if it hit you, versus a barely moving but very heavy rifle. Thus I would go with the lighter rifle hurting more.

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Hmmm... the shoulder ability would surely be of no consequence at such high speeds. It's definitely the lighter rifle then... –  YatharthROCK Jul 24 '13 at 4:10
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One way to think about this setup is to take the difference between the lighter rifle and the heavier rifle to the extreme.

Let's say you fix the heavier rifle very firmly to, say, a big slab of stone that is ground to give it very flat surfaces. Lift the slab+rifle onto an air table.

The layer of air provided by the air table will allow the slab to move with very little friction. So when you fire the rifle the recoil will set the slab+rifle in motion. The momentum of the slab+rifle will be equal to the momentum of the fired bullet; conservation of momentum. But the slab is far more massive than the bullet; its velocity will be low, and the motion of the slab towards your body will feel like being pushed very gently by an elephant.

By contrast, the lighter rifle will get a significant velocity from the recoil; when it hits your shoulder it feels like you've been struck by a hammer.

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10x for the analogy! –  YatharthROCK Jul 24 '13 at 4:08
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