Schrödinger's Cormorant

3.5 Finding the Energy

The relativistic definition of kinetic energy and the reason why anything that has mass cannot reach the speed of light.

When work is done on an object it makes it move, and the object therefore gains kinetic energy. Specifically, the work done is equal to the kinetic energy gained (where there is more than one operative force, it is the work done as a result of the net force that equals the accrued kinetic energy).

We can therefore use this general definition of work done (equation III-5) to derive the classical expression for the kinetic energy:

Question 3-10

The next question, then, is what about the relativistic expression for kinetic energy? We’ll take exactly the same approach as we did with the classical expression but this time you can look forward to a more challenging bit of integration.

Question 3-11

If you got through this derivation exercise: well done! It should, eventually, have led you to the correct relativistic definition for kinetic energy:

This is a remarkable expression: kinetic energy is expressed as the difference between two terms. Up till now – when we were looking at time, length and momentum – relativistic effects appeared just as simple “correction factors” to the familiar classical definitions. It was consequently easy to see that at velocities much lower than the speed of light these corrections were negligibly small and the classical expressions were therefore then valid. This is not so immediately obvious for the kinetic energy expression but we need to reassure ourselves that it is so:

Question 3-12

You should find, once again, that at low velocities, the classical and relativistic expressions will give more or less the same answer. The interesting question is what then happens as we start to get closer to the speed of light. Here's the answer:

As we've found with other quantities, that 1/√(1 – v2/c2) term really kicks in as we get close to light speed and in this case it means that the kinetic energy becomes much greater than classical theory would have suggested.

Question 3-13

This question provides an important lesson, not just for Cormorant but for us all. If you have mass, then reaching the speed of light would mean that you had infinite kinetic energy and this means that an infinite amount of work would need to be done to get you there. Ain't going to happen. So now we have a clear physical reason why the speed of light is unattainable for any object with mass.

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