[originally posted 2/17/08. Update added to end.]
Saturday, March 8, 2008
This post is in response to a comment on my post Physics Factoid: E = m c squared, which asks "Why is the conversion factor c2?" This will be slightly more technical than most of my posts, but should make sense to anyone who has had high school physics.
There are two parts to the question. First, why is the factor something squared? Second, why is that something c, the speed of light?
The answer to the first question is that the factor has to be some speed squared to get the units right. Most people don't value units enough. You can often solve a problem just knowing the units involved, and you can certainly tell if something is askew if the units aren't right. For example, if you ask someone, "what's the area of your living room?" and they answer, "Thirty feet," you know they misheard the question because an area has to be in square-feet.
The units of energy are mass·speed2, for example, kg·m2/s2. The kinetic energy of a mass m moving and velocity v is ½ m v2. [application to space junk]
So any formula writing energy in terms of mass has to involve something with units of speed squared.
But why the speed of light? Well, the theory of relativity is based on the idea that all observers, no matter how fast they are moving relative to you, must observe the same physical laws. And there is only one special speed which they all measure to be the same—the speed of light. In fact, you can show that the theory doesn't work if you try to add a second special speed. A short way of putting it is that in relativity there is only one constant to work with, the speed of light.
Now I could derive the formula for you, or show you it gives the correct kinetic energy in the low-speed limit, but a simple answer is that there is no other speed it could be!
So in summary, the factor has to be a speed squared to get the units right, and the only speed it could be is the speed of light.
[Answer to Comment added 3/8/08]:
Here is an attempt to answer the comment by 'dubiousraves'. I'm going to use some algebra, so stop reading now if you have math phobia! :)
I assume that you accept that the rest energy of any particle is some constant K times its mass, ie
rest E=K m
So we just need to figure out what K is. Further, its kinetic energy (at speeds much smaller than the speed of light) is ½ m v2. Now the total energy of a particle is its rest energy plus its kinetic energy. The theory of relativity says it has to be proportional to the factor called gamma, γ= 1/√(1-v2/c2), which is the same factor that accounts for the length contraction and time dilation. The total energy is thus
total E=γ K m.
When v=0, the particle is at rest and γ=1. In that case the total energy equals the rest energy (when something is at rest, it has no kinetic energy). Now when the speed v is a lot less than the speed of light (which is always true in everyday life), then v2/c2 is much smaller than 1. In that limit we can use the formula 1/√(1-e) ≈1+e/2 (this comes from a Taylor series expansion), for any quantity e that is much smaller than 1. So for small v, γ≈1+½ v2/c2 . If we plug this into the total energy, we get,
total E=γ K m≈(1+½ v2/c2) K m = Km + (K/c2) ½ m v2.
So to get the kinetic energy right, K needs to be c2. As a check, let's plug in K=c2:
rest Energy= K m = m c2,
total Energy = γ K m = γ m c2 ≈ (1+½ v2/c2) m c2 = mc2 + ½ m v2.
I hope that helps.