Lorentz Contraction with small velocity

1. Nov 24, 2008

benji55545

1. The problem statement, all variables and given/known data
About how many femtometers shorter than its rest length is the length of a car measured in the ground frame if the car is traveling at 30 m/s in that frame? Assume for the sake of argument that the car's rest length is 5.0 m. Remember that 1 fm = 10^-15 m.

2. Relevant equations
L=Lr*$$\sqrt{1-\beta^{2}}$$

3. The attempt at a solution

This is a pretty straight forward Lorentz contraction problem, but all attempts to actually calculate it fail because the velocity is so slow (1*10-7). How can I scale the numbers so solving 1-$$\beta^{2}$$ doesn't just give me 1?

Thanks for the help.

2. Nov 24, 2008

Chi Meson

It takes quite a bit of space (plus a masterful use of latex) to take you through the derivation of the "low velocity" formula. But it ends up as √(1-ß^2) ≈ 1-(0.5)ß^2.

The derivation involves going through a binomial expansion of the original formula. Do you recall doing this in class?

3. Nov 24, 2008

benji55545

Thanks for the reply. We talked about the binomial approximation when we were learning about about the relationship between spacetime and proper time, but I didn't think to apply it in this case.
Thanks for the help.