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Relativistic Mechanics

Problem 1.367

From the conditions of the foregoing problem determine the boost time τ0\tau_{0} in the reference frame fixed to the rocket. Remember that this time is defined by the formula τ0=∫0τ1−(v/c)2dt \tau_{0}=\int_{0}^{\tau} \sqrt{1-(v / c)^{2}} d t where dtd t is the time in the geocentric reference frame.

Reveal Answer
1.367. Taking into account that v=w′t/1+(w′t/c)2v=w^{\prime} t / \sqrt{1+\left(w^{\prime} t / c\right)^{2}}, we get τ0=∫0τdt1+(w′t/c)2=cw′ln⁡[w′τc+1+(w′τc)2]=3.5 months.  \tau_{0}=\int_{0}^{\tau} \frac{d t}{\sqrt{1+\left(w^{\prime} t / c\right)^{2}}}=\frac{c}{w^{\prime}} \ln \left[\frac{w^{\prime} \tau}{c}+\sqrt{1+\left(\frac{w^{\prime} \tau}{c}\right)^{2}}\right]=3.5 \text { months. }
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