# Does Newtons 3rd law mean that half of the energy is wasted?

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This may sound dumb to some folks, but lets say I fire a gun. 1000N of force goes into the bullet, and another 1000N of force goes into my body. Even if i strap the gun to the ground, it will just apply 1000N of force to move the earth by a tiny amount. Because of this, does it mean that it isn’t possible to have 2000N of force going to the bullet with the same parameters as before?

In: Physics

***Every*** reaction has an equal and opposite reaction.

No matter how powerful you make the gun, an ***exactly equal*** amount of force also happens in the opposite direction.

You can have the 2000N of force on the bullet, if you use more gun powder. No matter how you do it, the momentum of the bullet and the earth will be equal and opposite, mv=-MV, M=earth mass is huge, and V is tiny.  Both will have some energy imparted to them.  Bullet energy is mvv/2=MVv/2, and earth energy is MVV/2, which is much smaller and ignorable.

So force isn’t energy. They’re related, but not interchangeable. No, it’s not possible to have 2000N going into the bullet, because only 1000N of force was generated. That 1000N will press out on everything in all directions, but the important one is the direction the bullet travels. It’s also important to note that momentum is more of a factor here, not energy. You are correct, there is energy wasted. Not all the energy will go into the bullet. The recoil will waste some, some (a lot honestly) will become heat, and some will go into the exhaust gases. Some will even go into making the noise of the gun firing.

I’m not sure how much it’d be, but significant losses (more than half) wouldn’t surprise me.

No, 50% of the energy is not wasted.

Force does not equal energy. Force **times distance** equals energy. The gun moves a **lot** less than the bullet thus most of the energy goes into the bullet. Note that the distance moved I’m talking about is only the moment while the force is applied. The force only exists while the bullet is in the barrel (a very short time). Once the bullet leaves the barrel, the force is gone.

It’s not that half is wasted, it’s just a directionality thing.

For the gun example first of all the explosion of the powder in the bullet wants to expand outward in every direction but the design of the gun only allows the bullet to move in one direction, down the barrel.

If the same amount of powder was lit on an open surface with the bullet sitting on top it wouldn’t go nearly as far at all and in a fairly unpredictable direction.

You can’t have a single directional force though, you have to push off something. If the gun was redesigned that the bullet sat in the middle of a hollow tube with no chamber behind it the explosion wouldn’t propel the bullet very far (but farther than the open pile example) without the chamber to push against.

Newton 3 tells us that forces come in pairs (not apples). A force is a push or a pull. When you have a force it is the force of some object on some subject, due to some interaction. For example, if you throw a ball there is a force of you, on the ball, due to the contact between you and the ball.

For the Newton 3 pair, we switch around the subject and the object, switch around the direction, but keep everything else the same (same interaction).

So if you throw a ball, there is a force of you on the ball. There is an equal and opposite force of the ball on you.

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Energy is not force. Energy tells us how much something has been forced, and how much it could force other things. When looking at mechanics problems like these, we can look at them in terms of forces, or we can look at them in terms of energy (or we can look in terms of momentum). We get the same overall answer, but we’re using a slightly different set of tools to get there.

When you throw the ball you do work on it; you give it energy. You lose energy, the ball gains energy.

If we do this properly the maths gets rather complicated, but in simplest terms the work done by a force is given by the magnitude of the force, multiplied by the distance the thing being forced moves *in the direction of the force*.

So when you throw the ball, the force of you on the ball is in the direction you are moving the ball. So you do work on the ball; the ball gains energy.

But if we look at it the other way around, the force of the ball on you (the Newton3 pair) is in the opposite direction to the direction the ball is moving, meaning we get a negative sign in the energy; you *lose* energy, because you are doing work on the ball.

Newton3 gives us conservation of energy.

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Both ways of looking at this we only have one interaction. You are throwing the ball. But that interaction has a symmetry to it; if the ball gains energy, you lose energy. If you force the ball, the ball forces you. All we’re doing when splitting it up is looking at the two different sides to the interaction.

> 1000N of force goes into the bullet, and another 1000N of force goes into my body.

So applying it to this case; the gun applies a force of 1000N to the bullet. The bullet applies 1000N of force to the gun [with some other stuff going on, but we’re simplifying]. But that is only one interaction. We cannot somehow shift that to get 2000N to the bullet. In terms of energy, most of the energy will be going into the bullet, from the gun [from the chemicals that are exploding]. The gun loses energy, the bullet gains energy.

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As an aside; momentum is what happens when we sum our force over *time*, rather than distance moved (which gives us energy). So just as Newton3 gives us conservation of energy, if we take it into the energy world, Newton3 gives us conservation of momentum if we take it into the momentum world.

And if we’re really sneaky, once we get into working in 4-dimensions (with Special Relativity, and time being our 0th dimension), we may notice that energy is the time component, and momentum the space components, of the combined energy-momentum 4-dimensional thingamy.

Let’s break this down. A chemical explosion occurs, releasing heat, gases, and particulates. This mixture wants to occupy more volume than it has available, so pressure rises, exerting a force on all sides. The weakest point of the round fails and we now have a two body system. The bullet is pushed down the barrel. That same pushing force was on all sides of the casing, which is snugly held by the gun. The force moving the bullet is the same force acting on the gun. How things respond to pressure forces is based on mass, inertia. The bullet has very little mass compared to the sliding mechinism of a firearm. So that force acts to accelerate the slide back a little, while accelerating the bullet forwards a bunch. More energy transfer occurs to the lighter object relitive to the bigger object because of inertia. For a perfectly ideal situation, the ratio of energy transfer is the same as the ratio of masses. What Newton’s 3rd implies about this kind of system is that some energy is going to be “wasted” moving the bigger mass, but the bigger that mass is the less you have to dump into it. You can’t get anywhere without leaving something behind.

Every action has an equal but opposite reaction, so when a gun propels a bullet forward, the bullet exerts an equal amount of force on the gun. You can direct that force to some extent (such as where you vent the gas that’s released), but no, you won’t be able to put it into the bullet.

If you could magically redirect all that opposite force into the bullet, not only would you be able to fire the bullet twice as fast, you’d have a gun with zero recoil — something that every gun manufacturer would be very interested in, to put it mildly.

It doesn’t, and force doesn’t equal energy.

Consider Newtons Cradle- the ball and string thing you may have seen.

When one ball hits the other four, there’s a force to stop the moving ball, and this is transferred to an equal and opposite force that pushes the opposite ball. This battery continues, no waste, though friction and other forces do eventually slow the balls and stop the motion (“wasting the energy” – friction converts some of the energy to heat, the balls make some sound when they hit- the sound waves take energy)

Both the gun and the bullet are pushed with the same force.

But the bullet ends up moving a *lot* faster as it’s smaller.

The amount of energy an object has increases with the square of speed, so the smaller the bullet compared to the mass of the gun, the greater proportion of kinetic energy ends up in the bullet compared to the gun.

For an example that gives some nice round numbers.

M16 muzzle velocity is pretty close to 1000m/s with a bullet mass around 4g (0.004kg) – lot of variation by ammo type but those make nice working numbers.

That’s 4kgm/s of momentum (mass x velocity), and around 2000J of energy (0.5 x mass x velocity^2)

An M16 has a mass of around 4kg. So that 4kgm/s it receives due to Newton’s Third Law will move it at 1m/s. That’s 2J.

Now, not all the energy ends up in the round as there is wastage elsewhere, but in terms of efficiency from Newton’s Third Law, that gives you 99.9%. That’s one of a number of reasons why shooting a gun doesn’t kill you with the recoil.