Schrödinger's Cormorant

3.2 Don’t Lose Your Momentum

A reminder of the importance of the principle of conservation of momentum.

Cormorant is at a political meeting. Nefertiti is eavesdropping. "Our revolution is gathering momentum. Soon we will be unstoppable. United as seabirds, we will fly headlong into the armies of oppression and send them reeling into history!" After much, much more along similar lines, the meeting breaks up amid resolute squawking and flapping of wings. Cormorant and Nefertiti meet in the pub.

"Why do you use momentum as a metaphor in relation to your planned revolution?" she asks. "Keep your voice down!" hisses Cormorant. "Isn't it obvious? It is because momentum carries the implication that you can't stop it, moving relentlessly in the right direction, unless overwhelming force is brought to bear (in a unit length of time). And, of course, because it is the law of conservation of momentum that inspires us to believe that when the inevitable collision comes, with the walls of oppression that surround us, our accrued momentum will push those walls aside and those of who survive shall be free... I think it's your round, by the way."

Nefertiti is (quietly, slightly) impressed by Cormorant's grasp of the importance of momentum in physics. In an isolated system (with no force acting from outside) the total momentum inside the system is conserved, whatever the individual objects in the system may do. And, because momentum is a vector quantity, this applies individually in each of the three spatial dimensions. This is of massive importance because it is what makes it possible for us to calculate what will happen when collisions occur. It is one of the keys to predictability in the universe.

This sets Nefertiti wondering: does conservation of momentum work in the context of special relativity? If not, the entire universe would suddenly make no sense! She might as well join the seabird revolution. She must know ...

So, that's where we'll go next. Momentum may not be something you think about every day but it’s actually really important. We’ll be using it a lot, in fact, not just here but when we (eventually) get on to quantum theory too. So it’s important that we get into the world of momentum and address Nefertiti's worry. First of all, we’ll just pause for a moment to use Sir Isaac’s laws to remind ourselves about the connection between applied force and momentum change and then to derive the principle of conservation of momentum.

Question 3-1

We have an important relationship here, which we’ll make use of later:

Applied force, F = dp/dt

and if F = 0 then dp/dt = 0, which means that total momentum does not change.

Now here comes a Gedankenexperiment, to make sure you’re comfortable with this idea: you and Nefertiti enjoy hurling rocks at each other. Today you have two identical rocks (we’ll call them, imaginatively, rock A and rock B) that have been ground down to make perfect spheres with identical mass and smooth, frictionless surfaces, so that they can bounce off one another in a perfectly elastic way (meaning no kinetic energy is lost). You now each hurl your rock directly at the other, with identical speeds, so that the rocks collide head on, in mid air (so, happily, no blood is spilled). As no energy is dissipated in the collision, the rocks simply bounce off one another with their velocities exactly reversed.

Question 3-2

Now we come to the big question: suppose Cormorant is, as is his wont, flying overhead during such an incident: we know that his relative motion will mess with the velocities of the particles involved in the collision, so we have to re-open the question – does momentum still appear to be conserved in the collision, when observed from Cormorant’s moving perspective? That’s what we’re going to check out now – and a word of warning: Uncle Albert might be seriously upset if the answer is no because this is such a massively important principle, without which our ability to predict what happens when particles interact would be lost. We might as well be back in the stone age.

This is a more difficult question that you might think. Because, as we have seen, both time and space are affected by relative motion (time dilation and length contraction) we would need to derive a relativistic expression for velocity in order to calculate the momenta. We could do that but it’s not a quick job (look up Lorentz transformations in the Chapter 2 appendix, if you want to explore this further), and fortunately for us, Nefertiti has suggested a short-cut. She has appreciated that, since momentum is a vector quantity, it must be individually conserved in each perpendicular direction. This gives her the idea of a variant of the rock-throwing experiment that will allow you to focus on velocity perpendicular to the direction of relative motion of the two observers. This means there’s no length contraction to worry about, so it makes everything much simpler.

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