Inspired by this question here.
The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta \,d\phi^2).$$A tower has its base on the surface of this planet ($r = R$) and its top at radial coordinate $r = R_1$. A ball is held at rest by an observer at the top of the tower. It is then dropped and caught by an observer at the bottom of the tower.
What is the speed, $v$, of the ball as measured by the observer who catches the ball, just before the ball is caught?
Here, we are not assuming that $R \gg 2M$ or that $R_1 - R \ll R$. Also, I want the physical speed here, $v$, as would be measured, e.g. by a radar gun, not a coordinate speed, such as $dr/dt$.
Edit. This is not a homework question, and is merely inspired by the question I linked to, I am not sure why it is closed.
