Tuesday, April 9, 2013

Physics of Projectiles: What You Always Wanted to Learn in School

Here's what you've always wanted to know about physics but were never taught in school.

In an attempt to make things more interesting, I'll first state the conclusion, and then explain why it is so.

We'll consider a firearm bullet, but the same physics can be used for kinetic energy penetrators ("tank bullets"), asteroid impacts, rail guns, and pretty much any projectile that is not propelled by an engine. This excludes rocket propelled grenades and missiles.

Fact 1: The recoil you feel when firing your gun is exactly as big as the stagger that will be exerted on your target.

This comes from a conservation law, more precisely, conservation of momentum. Momentum is effectively "push" (the shooter calls it recoil). Momentum is mass times velocity (speed with direction).
If you double the mass of the bullet, recoil is twice as big. Likewise, if you double the speed of the bullet, recoil will be twice as big.

Conservation of momentum means that the push exerted on you by the bullet as it leaves your gun is the same as the recoil you will feel (the direction of the bullet's and your velocity have to be opposite):

bullet momentum = your momentum (recoil)

Which, according to what I just said, is:

(bullet mass) x (bullet velocity) = (your mass) x (your velocity)

A 9mm bullet weights approximately 9 grams. Muzzle velocity (speed upon exiting the muzzle of the gun) is about 365 meters per second. From this, we can calculate at what speed you would move backward when firing the gun:

(your velocity) = (bullet mass) x (bullet velocity) / (your mass)

If you weigh 75 kg, this gives a speed of 4 cm / second.

Assuming that the bullet doesn't slow down before hitting your target (which it will due to air resistance), the bullet's momentum will be the same upon impact, and again by the law of conservation, this momentum will be transferred to your target (assuming the bullet stops upon impact, otherwise only the proportion of the momentum that was lost is transferred).

Yes, this means that if your target weights 75 kg, it is going to be pushed back by precisely the same amount as the recoil you felt.

It also means that if your favorite bulletproof superhero is hit by a main tank gun, he's gonna fly back quite the distance through the skyline. The reason a tank doesn't fly back when hit is because it weighs a lot.

Momentum is not what causes damage upon impact. It is only the "push" exerted by the projectile.

Momentum = Recoil (or Stagger)

Technically, air resistance slows down a bullet, thus staggering the target even less than your recoil.

Fact 2: Transfer of Energy is what causes damage, mainly through conversion from kinetic energy (due to movement) into shock waves and heat release.

Energy, like momentum, can be derived from Newton's laws. Like momentum, kinetic energy is dependent on mass and speed, but the relationship is different:

The fact that Energy depends on the square of bullet velocity (velocity times itself) means the following:
when doubling the mass of the bullet, we double the kinetic energy (= the damage caused, as we will see soon). This is just the same as for momentum. But when doubling the speed of the bullet, we get four times more energy!

If we get ten times more mass, we get ten times more energy. But, again due to the square, if we get ten times more speed, we get 100 times (=10x10) more energy!

There is a law of conservation of energy. It is very fundamental, and it means that, assuming the bullet stops upon impact, all of the energy from the bullet will be delivered to the target. In practice, this means that the kinetic energy ("movement energy") from the bullet is entirely converted into shock waves and heat.

"Energy = Damage".

Technically, it's not really true that energy = damage, but (energy per time unit) = damage. If you convert nuclear fuel into electricity slowly, well, that's good cause it gives us heat in our homes. If you however release all that energy fast (nuclear detonation), then it's destructive. The difference being that in one case we release the energy over a long period of time, whereas in the latter we release a similar amount of energy over a very short period of time. However, all projectiles cause damage very fast, so this is not really an issue with bullets (as the example above makes clear, it does however matter when it comes to explosions).

Power = Energy per time = Damage.

Here's an interesting question: if the bullet's momentum is transferred to the shooter when firing, why isn't the energy also transferred to the shooter? The answer is that bullets are fired using a modern day version of gun powder, and the energy needed to speed up the bullet is taken from that powder (chemically stored energy).

Speeding up a bullet is thus done by converting energy stored inside the cartridge into kinetic energy. You however still feel recoil due to the bullet having mass (weight). It would in principle be possible to avoid that recoil by firing another bullet backwards at the same time, thus preserving momentum, but this is not done for obvious reasons. When (if?) if we develop handheld high energy firearms (e.g. lasers), we may get (almost entirely) recoil free weapons.

Fact 3: Firearms bullets are small and fast because that minimizes recoil while maximizing damage output.

But for now, we don't have high energy weapons, so we have an optimization problem (constraint problem) to solve:
Minimize recoil.
Maximize damage.

Which, after what we've said:
Minimize (bullet mass) x (bullet velocity) [Momentum]
Maximize (bullet mass) x (bullet velocity)^2 [Energy]

The solution should be obvious: since we much faster increase damage by increasing velocity (as opposed to mass), we should:
Minimize bullet mass.
Maximize bullet velocity.

There's one catch however: too fast traveling bullets means it could go right through the target, thus not delivery all of its kinetic energy onto the target, and risking to wound bystanders. When hunting large game, this is not usually an issue: a moose, being very thick, is likely to absorb all the kinetic energy. Some police forces uses hollow point bullets (see below), which decreases risk of firing through a target.

Ways to increase bullet mass include using larger calibers (and longer bullets) and very dense materials (such as tungsten or depleted uranium, not used in firearms). Increasing velocity includes using full metal jacket (pointy) bullets and rifling. More on these below.

Fact 4: Large Caliber = Bigger Hole = More Damage.

Sir Isaac Newton took us pretty far down the road, but there's more to the story. As you may know, caliber seems to be the single most mentioned aspect when it comes to firearms. In part this is because caliber constrains mass, but that's not all. A bigger hole causes more bleeding, which is obviously more damaging to organic targets. If you're in the forest facing a bear, the difference between a 9 mm bullet and a 12 gauge shotgun slug makes the difference between you being alive and dead.

Moreover, large calibers means more absorption of the kinetic energy upon impact, which in turns means more likelihood of not passing through (recall that if the bullet doesn't stop, not all energy will be delivered), and also that the time for transferring energy from kinetic to damage will be shorter: recall that damage = energy per second, so to maximize energy we need to transfer the energy as fast as possible.

Fact 6: Hollow Point bullets = More Damage. Pointy bullets = More Accuracy.

You've most likely seen pointy bullets in movies. Those are "standard", but not necessarily the best. Pointy means they penetrate the air better (i.e. they are less affected by air resistance). The downside, is that they can be too good at penetrating, going right through the target, thus not delivering all energy.

One solution is to use non-pointy bullets (i.e. bullets that are "hollow", so that it easily gets stuck in solids). This quickens the energy transfer (just as for large calibers above, but even more dramatically so), makes it more likely that the bullet will get stuck, but also decreases accuracy as it flies through the air less well.

A bullet type that attempts to take the best from both worlds is the plastic-tipped bullet: essentially a hollow point bullet with a plastic tip that easily falls off upon impact.

Fact 7: Rifling = More Accuracy.

The aerodynamics of a bullet can be significantly improved by imparting a spin on the bullet. This is due to the fact that bullets travel through air (sometimes water), and air is thick compared to the resistance free vacuum. Air resistance slows down the bullet, which means less kinetic energy (indeed, the energy is absorbed by the air). By spinning the bullet, it "cuts" through the air, much like a drill spins to penetrate wood or metal. Imparting spin on a bullet is achieved by simply rifling the barrel's inside.

1 comment:

  1. Quote: "It would in principle be possible to avoid that recoil by firing another bullet backwards at the same time, thus preserving momentum, but this is not done for obvious reasons."

    As a matter of fact, there actually are recoilless rifles [http://en.wikipedia.org/wiki/Recoilless_rifle]. For example the MBB Armbrust shoots shredded plastic with the same mass as its projectile backwards.