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LeconsdAnalyse's Channel
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Example► Let the object be a beam of light. So that ω=c. Find `u`.
Solution: u = (c + v)/(1 + c·v/c²)= c(1 + v/c)/(1 + v/c)= c □
Proof: From x=γ(x`+ v·t`) we get: dx=γ(dx`+ vdt`) -(1). From t=γ(t`+ v·x`/c²) we get: dt=γ(dt`+ v·dx`/c²) or, γ= dt/(dt`+ v·dx`/c²) -(2). Therefore, replacing `γ` in (1) with (2) we get: dx= dt(dx`+ vdt`)/(dt`+ v·dx`/c²) or, dx/dt= (dx`+ vdt`)/(dt`+ v·dx`/c²)= (dx`/dt`+ v)/(1 + v/c²·dx`/dt`). Set dx/dt=u, dx`/dt`=ω □