A planet is a uniform solid sphere of radius R=6400 km and surface gravity g0=10 m s^-2 (calculated as if it had no cavity). It contains a spherical…

Treat the cavity as a superposition of a full uniform sphere of density $\rho$ and a smaller sphere of density $-\rho$ placed at the cavity. At a point with position vector $\vec r$ from the planet's centre, the field of a full uniform sphere is $-\tfrac{4\pi}{3}G\rho\,\vec r$, and that of the negative-mass sphere is $+\tfrac{4\pi}{3}G\rho(\vec r-\vec d)$. Their sum inside the cavity is $-\tfrac{4\pi}{3}G\rho\,\vec d$: uniform, pointing towards the planet's centre. Since $g_0=\tfrac{4\pi}{3}G\rho R$, its magnitude is $g_0\dfrac dR=10\times\dfrac12=5\ \mathrm{m\,s^{-2}}$. The cavity diameter ($R$) far exceeds $10\ \mathrm{m}$, so the stone falls with constant acceleration: $h=\tfrac12g_ct^2$ gives $t=\sqrt{\dfrac{2\times10}{5}}=2\ \mathrm{s}$.