Answer
$$
\begin{aligned}
& v=-R \sqrt{\frac{2 g_0\left(y_0-y\right)}{(R+y)\left(R+y_0\right)}} \\
& v_{\text {imp }}=3.02 \mathrm{~km} / \mathrm{s}
\end{aligned}
$$
Work Step by Step
From Prob. 12-36,
$(+\uparrow) \quad a=-g_0 \frac{R^2}{(R+y)^2}$
Since $a d y=v d v$
then
$$
\begin{aligned}
& -g_0 R^2 \int_{y_0}^y \frac{d y}{(R+y)^2}=\int_0^v v d v \\
& g_0 R^2\left[\frac{1}{R+y}\right]_{y_0}^y=\frac{v^2}{2} \\
& g_0 R^2\left[\frac{1}{R+y}-\frac{1}{R+y_0}\right]=\frac{v^2}{2}
\end{aligned}
$$
Thus
$$
v=-R \sqrt{\frac{2 g_0\left(y_0-y\right)}{(R+y)\left(R+y_0\right)}}
$$
When $y_0=500 \mathrm{~km}, \quad y=0$,
$$
\begin{aligned}
& v=-6356\left(10^3\right) \sqrt{\frac{2(9.81)(500)\left(10^3\right)}{6356(6356+500)\left(10^6\right)}} \\
& v=-3016 \mathrm{~m} / \mathrm{s}=3.02 \mathrm{~km} / \mathrm{s} \downarrow
\end{aligned}
$$
