Square Frames of Reference

By | 2010-05-19

I have a physics problem. I last thought about it a few years ago, and thought I had forgotten it, but I found an old email where I described it. So that I don’t lose the idea again, I’m making it a blog post. (I’ve also got the answer I think).

Andy is riding his bicycle. Andy plus his bicycle weight, say 100kg. Andy accelerates from stand still to 10 metres per second. The energy required to do this is therefore:

E = 1/2 * m * v^2
= 0.5 * 100 * 10*10
= 5000 Joules

So far so good. Now, Andy takes his bicycle to a train station, gets on a train. Being a bad boy he decides to ride his bike on the train, while it’s moving (at say 100 metres per second). Now, he accelerates from stand still (on the train) to 10 metres per second (on the train); that is to say, that he accelerates from 100 metres per second to 110 metres per second (the train supplying the first 100 of course). The energy change for this is:

E = 1/2 * 100 * 110^2 – 1/2 * 100 * 100^2
= 605000 – 500000
= 105000 Joules

So; what’s the fault in the reasoning? On the train, Andy’s kinetic energy has increased by 105,000, but we know that Andy only supplied 5,000 joules of that so where is the extra 100,000 joules coming from?

As it happens, the answer to this one is easy. Newton’s third law tells us that for every action there is an equal and opposite reaction. In this case, the force required to accelerate the bicycle and rider also slows down the train — the train slows by (being enormous in comparison) an imperceptible amount — an amount sufficient to supply 100,000 additional Joules of energy.

Of course, this was merely the introduction. Now imagine that we aren’t talking about trains, we’re talking about unconnected objects in space. Unconnected is important, because I don’t want any of this equal-and-opposite stuff getting in the way.

The universe is empty, save for three 10kg bowling balls. Somehow (we’ll ignore how) they are moving in the same direction but at three different speeds. Of course, being that they are in space alone, there is nothing we can measure their speeds against except each other. So, let’s say this:

  • Ball2 is moving 10 metres per second faster than Ball1
  • Ball3 is moving 10 metres per second faster than Ball2

Let’s begin by using Ball 1 as our frame of reference, and add up the total kinetic energy in the universe. From Ball1’s point of view, Ball2 is travelling away from it at 10 metres per second, and Ball3 is travelling away from it at 20 metres per second…

  • E1 = 0
  • E2 = 1/2 * 10 * 10^2 = 500 Joules
  • E3 = 1/2 * 10 * 20^2 = 2,000 Joules

Fine, total kinetic energy, 2,500 Joules. Let’s go again, but using Ball 2 as our frame of reference. From Ball2’s point of view, Ball1 is travelling away from it at 10 metres per second, and Ball3 is travelling (in the opposite direction) away from it at 10 metres per second.

  • E1 = 1/2 * 10 * 10^2 = 500 Joules
  • E2 = 0
  • E3 = 1/2 * 10 * 10^2 = 500 Joules

Total kinetic energy, 1,000 Joules. What? The first law of thermodynamics tells us that energy can be neither created not destroyed, and yet in a universe that has had nothing changed, just by looking at it from a different point of view we’ve lost 1,500 Joules of energy.

That can’t be: there must be a fixed amount of energy in the universe.

This is where I got stuck the last time — where did the energy go? Of course the answer is “nowhere”, the laws of thermodynamics are as close as we get to inviolable in science. Sir Arthur Stanley Eddington once wrote:

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

The answer turns out to be hidden in a thing called the centre of momentum frame. It’s officially that frame (or rather one of the infinite number) in which the centre of mass of a system is at rest. In this frame, the total kinetic energy is called the rest energy of the system.

From the analysis above, we find it anomalous that kinetic energy is not invariant between reference frames — rather than being a mistake however, it turns out that it is true. Let’s work it out for our example above…

First, the velocity of the centre of momentum frame. Since momentum is linear with speed and mass, it doesn’t matter which frame we pick to measure speeds, we will always come up with the same total momentum, we’ll use Ball1 as our reference frame for calculating momentum:

p = m1 * v1 + m2 * v2 + m3 * v3
= 10 * 0 + 10 * 10 + 10 * 20
= 300 kgm/s

This is relative to our reference frame of course. Let’s turn this into a velocity:

p = mv
v = p / m
= 300 / (10 + 10 + 10)
= 10 m/s

That is to say, that the centre of momentum frame travels at 10 m/s relative to Ball1. Not coincidentally, that’s exactly the same as Ball2 (as we’d expect, as the system is symmetric). We have already calculated the total kinetic energy in that frame, 1,000 Joules. This is the rest energy of this system.

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