Question

(a) Find the value of $\gamma$ for the following situation. An
astronaut measures the length of her spaceship to be 25.0 m, while an Earth-bound observer measures it to be 100 m. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Final Answer

- $0.250$
- $\gamma$ must always be greater than 1.
- It's unreasonable to think the Earth based observer would measure a greater length.

### Solution video

# OpenStax College Physics, Chapter 28, Problem 18 (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.