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A cannonball firing from a cannon is often given as an example of Newton's third law. The explanation goes like this: The cannon exerts a force on the cannonball and thus the cannonball exerts an equal but opposite force on the cannon. The ball accelerates rapidly and the cannon recoils in the opposite direction, but with much less acceleration than the cannonball because the cannon is much heavier (Newton's second law).

I am confused by this explanation. The cannon is fired when an explosive charge is detonated, causing a sudden and immense increase in pressure. Is it not this pressure that causes the rapid acceleration of the cannonball and the recoil of the cannon, not an action–reaction between the cannon and the cannonball? If no cannonball is present when the charge is detonated, then the pressure dissipates much more quickly and the recoil is smaller but still present (at least going by this question on Quora), which seems to also go against the action–reaction explanation.

Or am I talking nonsense? I suspect – or rather hope – that the above paragraph is correct but incomplete. I am sure Newton's third law (or the conservation of momentum) plays a fundamental role, but I am struggling to come up with a satisfactory explanation.

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The cannon is fired when an explosive charge is detonated, causing a sudden and immense increase in pressure. Is it not this pressure that causes the rapid acceleration of the cannonball and the recoil of the cannon, not an action–reaction between the cannon and the cannonball?

For purposes of this model, we can consider the expanding gas from the explosion to be part of the cannon, or as an intervening object between the cannon and the ball. So the gasses exert a force on the ball. The ball exerts a force back on the gasses. This is transferred to the cannon.

You could also imagine or build a (toy) "cannon" with a spring mechanism to propel the ball, rather than an explosion. You'd see very similar results.

In any case, the deeper point, which you will soon learn, is that momentum is a conserved quantity. Regardless of what mechanism applies the force on the ball and the cannon, after the ball is flying free the cannon must end up with as much backwards momentum as the ball has forward momentum.

If no cannonball is present when the charge is detonated, then the pressure dissipates much more quickly and the recoil is smaller but still present

Because air and exhaust gasses from the explosion are expelled from the cannon. These gasses have mass and carry momentum, therefore they exert a reaction force on the cannon just as a ball does.

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  • $\begingroup$ Is the point that since the cannon–cannonball system started off with zero momentum, its net momentum must remain zero after the detonation, i.e. the momentum of the cannon and the momentum of the cannonball must sum to zero? $\endgroup$ – dwolfeu Nov 18 at 21:28
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    $\begingroup$ @dwolfeu, yes, assuming you're neglecting friction and other nonidealities. $\endgroup$ – The Photon Nov 18 at 21:54
  • $\begingroup$ Well, if you don't include the gas as part of the cannon, then it can carry momentum separate from the ball and cannon. $\endgroup$ – Acccumulation Nov 19 at 5:10
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    $\begingroup$ @Acccumulation, I classify that under "other nonidealities". $\endgroup$ – The Photon Nov 19 at 5:21
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You have stumbled on physics education's ugly secret: we tell students convenient lies. Not really. We simply don't tell all the truth, and we build simplified models so that students can learn basic skills and principles.

The mass of the gas is very small compared to the masses of the cannon and the projectile, so we model the system as an interaction between the cannon and the projectile. The momentum given to the gas and unburned fuel is small, but not zero, as you pointed out.

Your question raises an issue: If the gas is expanding with the projectile interface moving faster than the cannon interface, then the center of mass of the gas is moving "forward". That means the force the projectile exerts on the gas is smaller than the force the cannon exerts on the gas. This means that our simple model (force on cannon = force on projectile) is faulty/incomplete. I would say incomplete, because in reality, it's the gas which interacts with the projectile, and we ignore the gas.

It also means that there is a pressure differential across the gas, which is not surprising. Burning fuel is not a quasi-static system.

Momentum is conserved, but in our analysis and teaching we sometimes ignore small details like "what causes the forces that accelerate the projectile and the cannon." In reality, the whole balance of powder load, projectile mass, burn rate, cartridge length, etc, is extremely complicated. That's why we do experiments!

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  • $\begingroup$ What??  Next, you'll be telling me there's no such thing as a spherical elephant in a vacuum… $\endgroup$ – gidds Nov 19 at 15:47
  • $\begingroup$ @gidds It only exists on the flat, non-rotating, airless Earth. $\endgroup$ – Bill N Nov 19 at 15:50
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In retrospect Newton's third law asserts isotropy of space.

I noticed many authors of textbooks opting for that reinterpretation. What that does is that it moves the third law away from the realm of causality

Causality
Causality is about how things happen sequentally in time. Causality is is all about time.

On the other hand, asserting that space is symmetric is complementary to any assertion about time.


The cannonball being propelled out of the cannon and the recoil of the cannon happen simultaneously: neither is the cause of the other.


About assertions of isotropy of space:
Newton's first law can be reinterpreted as asserting that space is euclidean, for motion of objects that are not subject to any force. More precisely: we can reinterpret the first law as asserting that space has the same symmetries as the geometric concept of euclidean space. (Symmetry under translation, symmetry under orientation change.)

The third law also asserts the isotropy of space, for pairs of objects that are exerting a force upon each other.


It's tempting to speculate why Newton felt a need for the third law.
My hypothesis for that starts with the point that to make the celestial mechanics work Newton needed that gravitional mass is equal to inertial mass.

But that makes gravity a rather nebulous force. (Earth is in orbit around the Sun, but we don't feel the gravitational attraction from the Sun because gravitational mass is equal to inertial mass.)

In Newton's time the observatoinal data were sufficiently accurate to demonstrate that the Sun and Jupiter are both orbiting their common center of mass. That is, the motion of the Sun and Jupiter demonstrates that the Sun and Jupiter, exerting gravitational force upon each other, are behaving as a third law force pair.

If the Sun would be stationary, despite the burden of having to swing Jupiter around, that would be very bad news for newtonian theory of gravity.

But: the Sun isn't stationary! The Sun and Jupiter's motion is in accordance with the motion of a third law force pair. Thus it is justified to regard gravity aa a force.

That is my hypothesis as to why Newton felt a need for the third law.

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