What Velocity is Needed for a Rocket Pilot to Reach a Star in One Year?

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SUMMARY

The discussion focuses on calculating the velocity required for a rocket pilot to reach a star 4 light years away within one year of the pilot's time, utilizing concepts from special relativity. The Lorentz factor is essential for understanding the relationship between proper time and distance as perceived from different reference frames. The key formula discussed is T' = d'/v, where T' represents the proper time, d' is the distance in the pilot's frame, and v is the velocity. The participant seeks clarification on the transition between specific lines in a provided solution, indicating a need for deeper understanding of relativistic equations.

PREREQUISITES
  • Understanding of special relativity concepts, including time dilation and length contraction.
  • Familiarity with the Lorentz factor and its application in relativistic physics.
  • Basic knowledge of algebra and physics equations related to motion.
  • Ability to interpret and manipulate equations involving proper time and distance.
NEXT STEPS
  • Study the derivation and implications of the Lorentz factor in special relativity.
  • Learn how to apply the time dilation formula in various scenarios.
  • Explore the concept of proper time versus coordinate time in different reference frames.
  • Review examples of relativistic velocity calculations in physics literature.
USEFUL FOR

This discussion is beneficial for physics students, educators, and anyone interested in the practical applications of special relativity in astrophysics and rocket science.

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Homework Statement



The pilot of a rocket wants to reach a star in 1 year. The rocket is currently on Earth. The star is 4 light years away, measured from an observer on Earth. What velocity is needed for the pilot to reach the star in a year in the pilot's/rocket's time?

Homework Equations



See below. Relativistic Length = Proper Length / Lorentz factor is used below, to make the substitution from Line 2 to Line 3 of L to L'/(Lorentz factor). The gamma symbol indicates the Lorentz factor.

The Attempt at a Solution



I have been given a solution, I just don't understand it. Is someone able to explain the jump from lines 6 to 7 in this solution? http://i.imgur.com/F70Oy.jpg . T' and L' indicate the proper time (time is from pilot's perspective, hence 1 year, distance from Earth's perspective, hence 4 light years). Tp is used the in first line, it is the same as T', hence where the T' comes from in Line 4.

I have been struggling with this for some time. Any help is appreciated, thank you!

Edit: I believe there is a substitution of a formula from this section of Wikipedia but I am unsure which one. Also, Wikipedia uses different symbols (in each formula, variables without a dash indicate measurements made in the stationary reference frame, unlike mine). https://en.wikipedia.org/wiki/Special_relativity#Time_dilation_and_length_contraction
 
Last edited:
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Why not just do it yourself? I'll get you started

t'=d'/v
 

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