The displacement $x$ of a particle of rest mass $m$, resulting from a constant force $mg$ along the $x$-axis, is $$x = \frac{c^2}{g}\left[\sqrt{1+\left(g \frac{t}{c}\right)^2}-1\right].$$

Approximate $x$ as a function of time $t$ with the first two nonzero terms of the Taylor polynomial:
$x\approx$ .

(Note: Newtonian physics models this situation with the familiar $x=\frac{1}{2}gt^2$.)