Question

If relativistic effects are to be less than 3%, then γ must be less than 1.03. At what relative velocity is $\gamma = 1.03$?

Final Answer

$7.183\times 10^{7}\textrm{ m/s}$

### Solution video

# OpenStax College Physics, Chapter 28, Problem 8 (Problems & Exercises)

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Video Transcript

This is College Physics Answers with Shaun Dychko. What is the relative speed when the Lorentz factor—

*γ*—is 1.03? So*γ*is 1 over the square root of 1 minus the relative speed squared over*c*squared and we can multiply both sides by 1 minus*v squared*over*c squared*and divide both sides by*γ*and we get square root 1 minus*v squared*over*c squared*is 1 over*γ*and then we can square both sides and get 1 minus*v squared*over*c*squared is 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 we get*v squared*over*c squared*is 1 minus 1 over*γ squared*then multiply both sides by*c squared*and you get*v squared*is*c squared*times 1 minus 1 over*γ squared*then square root both sides and finally we have a formula for the speed*v*in terms of speed of light and*γ*so we have speed of light times square root 1 minus 1 over*γ squared*. So that's 2.998 times 10 to the 8 meters per second times square root 1 minus 1 over 1.03 squared and that's 7.183 times 10 to the 7 meters per second. So at speeds greater than this, you will have a Lorentz factor that is creating a relativistic effect that exceeds 3 percent.