Question
(a) Find the value of γ\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?
Question by OpenStax is licensed under CC BY 4.0
Final Answer
  1. 0.2500.250
  2. γ\gamma must always be greater than 1.
  3. It's unreasonable to think the Earth based observer would measure a greater length.

Solution video

OpenStax College Physics for AP® Courses, Chapter 28, Problem 18 (Problems & Exercises)

OpenStax College Physics, Chapter 28, Problem 18 (PE) video thumbnail

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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.