[Physics] How to calculate stopping power

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Malcolm Gladwell made a claim in a recent talk that a sling with a stone going at 30m/s has the same stopping power as a .45 calibre handgun.

How would I verify whether or not this claim is true – even given some assumptions, like the stone has a weight of 1lb or 2 lbs or whatever.

Not quite sure how to work this out.

Assume that the bullet is a 185grain 0.45 calibre bullet.

Best Answer

While I agree with the caveats made by dmckee in his comments, there is an obvious interpretation of stopping power as the change in momentum caused by the projectile.

Collision

The mass and velocity of the projectile are $m$ and $v$ respectively, and the mass of the target is $M$. Since the target is stationary the initial momentum is just $mv$.

Assuming the bullet transfers all its energy to the target (i.e. it stops within it and doesn't go through and out the other side) then after the impact the projectile and the target will be moving with some velocity $v'$, so the total momentum is $(m + M)v'$. Conservation of momentum tells us that the initial and final momenta must be the same so:

$$ (m + M) v' = mv $$

and rearranging this gives:

$$ v' = \frac{m}{m + M} v $$

Let's say the stone weighs 0.1kg. My mass is about 70kg, so if the stone hit me then final velocity would be:

$$ v' = \frac{0.1}{0.1 + 70} 30 \approx 0.043 m/s $$

Now for the bullet. I guess it depends on which pistol you choose, but the .45 ACP has a muzzle velocity of about 350 m/sec and a 185 grain bullet weighs about 0.012kg so:

$$ v' = \frac{0.012}{0.012 + 70} 350 \approx 0.06 m/s $$

which is actually pretty close to the stone.

Really the key bit of the calculation is the initial momentum of the projectile, $mv$. The stone weighs about ten times as much as the bullet but is moving at about one tenth of the speed, so the two factors of ten cancel out and the initial momentum is roughly the same. This is presumably the basis of the claim about stopping power. But attend to dmckee's comments. The velocity changes calculated above are around 0.04 m/s, and since humans can run at several metres per second you're not literally going to stop a running human with either the gun or the sling. Presumably a gun stops someone because it makes them fall, and friction with the ground does the stopping, in which case you should probably avoid trying to shoot axe wielding maniacs when standing on an ice rink.

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