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## Homework Statement

In the twins paradox, suppose that Florence begins at rest beside Methuselah, then accelerates in Methuselah’s x-direction with an acceleration a equal to one Earth gravity, “1g”, for a time ##T_F/4## as measured by her, then accelerates in the −x-direction at 1g for a time ##T_F/2## thereby reversing her motion, and then accelerates in the +x-direction at 1g for a time ##T_F/4## thereby returning to rest beside Methuselah. Show that the total time lapse as measured by Methuselah is:

$$T_M=\frac{4}{g}\sinh{(\frac{gT_F}{4})}$$

## Homework Equations

(1) ##T_F=\int d\tau = \int\sqrt{dt^2-\delta_{ij}dx^idx^j}=\int_0^{T_M}\sqrt{1-v^2}dt## where ##d\tau## is proper time of Florence

## The Attempt at a Solution

I tried to integrate (1) with ##v=gt## but I have got some crazy result with ArcSin and square root. It is impossible to extract ##T_M##.

Is my procedure generally wrong? Or it exists some better way how get result for this case?