All Problems Relativistic Mechanics
Problem 1.367
From the conditions of the foregoing problem determine the boost time \(\tau_{0}\) in the reference frame fixed to the rocket. Remember that this time is defined by the formula \[ \tau_{0}=\int_{0}^{\tau} \sqrt{1-(v / c)^{2}} d t \] where \(d t\) is the time in the geocentric reference frame.
1.367. Taking into account that \(v=w^{\prime} t / \sqrt{1+\left(w^{\prime} t / c\right)^{2}}\), we get \[ \tau_{0}=\int_{0}^{\tau} \frac{d t}{\sqrt{1+\left(w^{\prime} t / c\right)^{2}}}=\frac{c}{w^{\prime}} \ln \left[\frac{w^{\prime} \tau}{c}+\sqrt{1+\left(\frac{w^{\prime} \tau}{c}\right)^{2}}\right]=3.5 \text { months. } \]