Resultant Gamma after Einstein velocity addition

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SUMMARY

The discussion centers on the application of the Einstein velocity addition formula in the context of relativistic physics, specifically regarding the resultant gamma factor when a projectile is fired from a moving rocket. The participants seek a compact equation for the resultant gamma (\(\Gamma'\)) expressed solely in terms of the original gammas (\(\gamma\) for the rocket's velocity \(v\) and \(\Gamma\) for the projectile's velocity \(u\)). The conversation highlights the challenge of deriving this formula and references the Relativistic Doppler Factor as a potential parallel, although it is not the exact solution sought.

PREREQUISITES
  • Understanding of Einstein's theory of relativity
  • Familiarity with the Einstein velocity addition formula
  • Knowledge of gamma factors (\(\gamma\) and \(\Gamma\)) in relativistic contexts
  • Basic concepts of the Relativistic Doppler Effect
NEXT STEPS
  • Research the derivation of the Einstein velocity addition formula
  • Explore the mathematical properties of gamma factors in special relativity
  • Study the Relativistic Doppler Effect and its applications
  • Investigate existing literature on compact equations for resultant velocities in relativity
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Students and professionals in physics, particularly those focusing on special relativity, as well as educators seeking to clarify concepts related to velocity addition and gamma factors.

wil3
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Hello. Let's say that I am in a frame in which I see a rocket traveling at v. This rocket then fires a projectile forward with velocity u in its rest frame.

I can find u' easily enough using the Einstein velocity addition formula. However, I recall seeing a version of the formula that uses gammas instead(ie, if \gamma is defined for v and \Gamma is defined for u, then there is some compact equation for \Gamma' purely as a function of the other two gammas)

Does anyone know this formula? It is proving very difficult to derive. Thanks very much.
 
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Could you be thinking of the Relativistic Doppler Factor where you can just multiply them for multiple velocities?
 
Nah, I remember this came up in class on a problem of this exact sort. I seem to remember there being a \Gamma*\gamma term going on somewhere. But thanks for the analysis-- I'm sure there is a parallel with the Doppler Effect.
 

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