It’s the distribution of the force.

All that force is coming down on a single point when the bullet hits.

Like pushing in a drawing pin, the thin end goes into the wall and the thick end doesn’t hurt your finger because the same force is distributed over a larger area.

Is the force not also distributed using a bullet proof vest?

It's not the same force.

Energy = force * distance(in the direction of motion), and momentum = force * time.

There's much less distance and time during the impact than there is when firing the projectile, so force is greater.

Yes. But you are also talking about a LOT more energy, as noted elsewhere in this thread.

You're missing the fact that the recoil is already distributed over a larger area when it impacts the vest (the butt of the weapon has a much larger impact area than the point of the bullet)

This is correct.

To add on to it, same reason why being hit in the hand with a baseball line drive will probably break your hand, but you're fine swinging the bat.

The force is distributed over a larger area - less force is applied on any given area, so it does less damage

The force is spread over a longer period of time so the amount of force at any given time is going to be lower.

Similarly with a gun, the recoil is going to be in the neighborhood of 0.45m/s. The bullet is going to be traveling at 370m/s. The bullet is going to apply all of its force hundreds of times faster than the gun, on a dramatically smaller surface area.

The effect comes from the pressure exerted: the force divided by the area it acts over. The tip of a bullet has a very low area thus a very high pressure because all the force acts through a single point. The recoil is an equivalent force but the gun is held in one or both hands so the force pushes up the arms to the body and acts against your whole body mass, thus dissipating its effect.

It is, and that's one of the ways the vest protects you, but it can't absorb all of the force and that's why you get a bruise instead of hole in you.

One thing is that a bullet, when fired, accelerates down the barrel over time, applying a lower force over time. However, when it hits a hard target direct on, the velocity comes to an immediate stop, which means a suddenly negative acceleration, jerk, and very high force (F=m*a). This is why hollow points tend to have better stopping power than FMJ that go right thru a person.

Also, the recoil of shooting a high caliber handgun or rifle does leave some soreness in the hands/shoulder.

It’s not about force being spread over a larger area.

There are two things here:

1. Momentum is conserved. Momentum is the mass of an object times it’s velocity. Velocity has a direction. The total momentum of gun+bullet before you shoot is zero, so it has to be zero after you shoot. So when you shoot the bullet speed times bullet mass in one direction has to equal the gun speed times the gun mass in the other direction. But the gun is much much heavier. To keep math easy we’ll say 100x heavier. So for the mass of the bullet times the velocity of the bullet(plus propellant) to have the same magnitude of momentum as the gun the bullet has to go one way 100x faster than the gun goes the other.

2. The kinetic energy is proportional to the mass times the velocity squared (1/2 mv^2). The bullet weighs 1/100 times that of the gun, but is going 100x faster (to keep numbers easy). This means that the bullet is carrying 100x as much kinetic energy as the gun.

So for every unit of energy your hand/shoulder has to absorb the bullet proof vest (combined with your flesh/bones behind it) need to absorb 100 units, which is gonna take a lot more to absorb. With something like a rifle, much heavier than the bullets it fires, that could go to 200x, 300x or more.

There are two relevant concepts in physics at play: Momentum and (kinetic) energy.

When you fire the bullet:

• Both the bullet and the shooter get an equal (but opposite) amount of momentum. (see conservation of momentum).
• Momentum = mass x speed. If the mass of the person is 100kg and the bullet weighs 50g the ratio of speed difference between the bullet and the shooter is 50 gr / 100kg*100 = 1/200. This is important for the energy part.
• Both the shooter and bullet get kinetic (movement) energy, but in different amounts. The energy is 0.5 x mass x speed x speed. As per above, the mass and speed of the shooter and bullet are related to each other. As you can calculate, the speed increase of the person is much lower which results in not much energy being transferred to the person.
• In essence, most of energy goes to the bullet since the ratio of weight of the person to the bullet is very large. All kinetic energy is coming from the gun powder (conservation of energy).
• You can imagine receiving the bullet must result in the bullet proof vest to absorb a crazy amount of energy (200 times more than receiving the recoil).

To sum it up, when firing a bullet (on a slippery surface):

• Both the bullet and the shooter are pushed by the explosion, the bullet will speed up much faster as it is lighter.
• The speed difference is by a factor of (weight shooter / weight bullet) and generally > 200.
• The explosion energy is put into both the shooter and the bullet. The energy is split unevenly and favors the fast bullet (by again the ratio of weight shooter / weight bullet). The bullet receives most energy, the shooter does not receive much energy in the recoil.
• Receiving the bullet in the bullet proof vest requires >200 times as much energy absorption compared to the recoil of shooting a bullet.

Also known as pressure.

It would be like holding the gun at your chest. Pretty painful.

Well, you are missing a couple of things.

#1. The force expended will take the path of least resistance. Which is pushing the bullet out and down the barrel. At which point a portion of the force it released out the end with not resistance.

#2. Concentrated above force is behind a small object. That has little to no resistance and is magnified by all force being directed from behind to push it out of the barrel.

#3. The recoil is only the force that is not expended out of the barrel.

#4. If you plug the barrel, you have violent explosions where the force has no place to release and often kills the person holding the gun because the force is directed equally in all directions.

If you had a firecracker and placed, it on your open palm and lit it. It exploded it would hurt like hell but you might not have massive damage. Lit it and close your hand around with a fist and you are going to loose fingers. The blast from the firecracker dissipates in all directions and you hand only gets a portion of that expelled blast. Close your hand and you hand takes it all. Sort of the same idea.

TRDL: The vast majority of the force produced is concentrated and expelled out of the barrel where there is no resistance. In recoil you are only getting the force that could not be vented out the barrel and in the vest getting hit by the bullet you are getting everything else which is the vast majority of the expended blast translated into kinetic energy.

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For some reason, no one else has mentioned that a person firing a gun knows that the recoil will occur when they pull the trigger and will brace themselves against that force, while taking a round into a bulletproof vest is usually a surprising event that the subject hasn't really prepared for.

It's akin to a person in a physical confrontation might easily brace against an obvious shove from someone in front of them, while an identical shove coming suddenly behind would knock them over.

That’s a great detail! I’m going to expand on it in case OP reads here

Gunpowder deflagrates instead of detonating. It burns instead of booms. Well, it does boom from our perspective, but if you watch in slow motion it takes a lot more time than dynamite. This gradually increases the gas pressure to start the bullet moving.

Bullets fit very tightly with the barrel. Imagine a tennis ball stuck tightly into the end of a pipe. You could punch it with all your strength, but it wouldn’t go very far and you’d probably damage yourself, the pipe and the ball. If you start pushing lightly and get it moving and then push harder, you’ll expend the same amount of energy and get it much farther down the pipe without damaging anything.

This is also why some calibers benefit from a longer barrel. If the bullet leaves the barrel before all the powder finishes burning, it will miss out on some velocity. Some calibers are designed to burn quickly so barrels can be short, but they also need to be designed strong to handle the higher peak pressure.

On top of all that, if the gun uses some of that recoil energy to do something useful like load another round, the impulse will be even more spread out on the shooter.