Originally Posted by
crowhill
If you accept that the rest energy of any particle is some constant K times its mass, ie
rest E=K m
we need to figure out what K is. 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 , 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.
Therefore c has to be c squared.
Re: stuck
do more work
eat a lot less
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