Lorentz Contraction with small velocity

[benji55545]

Homework Statement

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.

Homework Equations

L=L_r*$$\sqrt{1-\beta^{2}}$$

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.

 

[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?

 

[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.