Question

(a) At what relative velocity is $\gamma = 2.00$? (b) At what relative velocity is $\gamma = 10.0$?

Final Answer

- $2.60\times 10^{8}\textrm{ m/s}$
- $2.98\times 10^{8}\textrm{ m/s}$

### Solution video

# OpenStax College Physics, Chapter 28, Problem 10 (Problems & Exercises)

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Video Transcript

This is College Physics Answers with Shaun Dychko. At what relative velocity will the Lorentz factor

*γ*be 2.00? That's part (a) and in part (b), we'll say what speed would give a Lorentz factor of 10? So*γ*is 1 over the square root of 1 minus*v squared*over*c squared*, we are going to do some algebra to solve for*v*. First, we'll multiply both sides by 1 minus*v squared*over*c squared*and divide both sides by*γ*. So that gives us square root 1 minus*v squared*over*c squared*equals 1 over*γ*. Then we'll square both sides that gives us 1 minus*v squared*over*c squared*equals 1 over*γ squared*and then add*v squared*over*c squared*to both sides and subtract 1 over*γ squared*from both sides and you get*v squared*over*c squared*is 1 minus 1 over*γ squared*after switching the sides around. Then multiply both sides by*c squared*and you get this line here and then square root both sides and*v*then is the speed of light times the square root of 1 minus 1 over*γ squared*. That's 2.998 times 10 to the 8 meters per second times the square root of 1 minus 1 over 2.00 squared and that's 2.60 times 10 to the 8 meters per second. We can repurpose this same formula here for part (b) and we'll just plug in numbers so we have the speed*v*in part (b) when*γ*is 10.0 is going to be 2.98 times 10 to the 8 meters per second.