- #1

- 668

- 68

## Homework Statement

Assume a particle moves along the x-axis. It is uniformly accelerated, in the sense that the acceleration measured in its instantaneous rest frame is always g, a constant. Find x and t as functions of the proper time tau if the particle is at x0 at t=0 with no velocity.

## The Attempt at a Solution

I am assuming the wording of the problem means

[tex]\frac{d^2 x}{d \tau^2}=g[/tex]

which after some work gives the following results:

[tex] x(\tau) = \frac{1}{2}g \tau^2+x_0[/tex]

[tex]\frac{dx}{d \tau} = \frac{dx}{dt} \frac{dt}{d \tau} = g \tau [/tex]

[tex]\frac{dx}{dt} = \frac{g \tau}{\sqrt{1+(g \tau)^2}}[/tex]

[tex] \frac{dt}{d \tau} = \sqrt{1+(g \tau)^2} \implies t(\tau) = \frac{g \tau \sqrt{1+(g \tau)^2)}+sinh^{-1}(g \tau)}{2g} [/tex]

Is this correct? This expression for the time as a function of tau seems bizarre for such a simple situation, and it makes me wonder about whether my interpretation of the problem wording was correct.

Last edited: