Chapter 2.3, Problem 28P

(a) Suppose that a body is dropped $\left({\upsilon }_0=0\right)$ from a distance $r_0>R$ from the earth’s center, so its acceleration is $d\upsilon /dt=-GM/r^2$. Ignoring air resistance, show that it reaches the height $r<r_0$ at time $t=\sqrt{\frac{r_0}{2GM}}\left(\sqrt{r{r}_0-r^2}+r_0{\cos }^{-1}\sqrt{\frac{r}{r_0}}\right)$ .

(Suggestion: Substitute $r=r_0{\cos }^2\theta$ to evaluate $\int \sqrt{r/\left(r_0-r\right)}dr$ .) (b) If a body is dropped from a height of 1000 km above the earth’s surface and air resistance is neglected, how long does it take to fall and with what speed will it strike the earth’s surface?