Answer
(a) $U(x) = \frac{(0.800~N~m^2)}{(x+0.200~m)}$
(b) $v = 3.27~m/s$
Work Step by Step
(a) $U(x) = -\int_{\infty}^{x}F_x~dx$
$U(x) = -\int_{\infty}^{x}\frac{\alpha}{(x+x_0)^2}~dx$
$U(x) = \frac{\alpha}{(x+x_0)}\vert_{\infty}^{x}$
$U(x) = \frac{\alpha}{(x+x_0)}$
$U(x) = \frac{(0.800~N~m^2)}{(x+0.200~m)}$
(b) $K_2+U_2=K_1+U_1$
$K_2 =0+U_1-U_2$
$\frac{1}{2}mv^2 = U_1-U_2$
$v^2 = \frac{2U_1-2U_2}{m}$
$v = \sqrt{\frac{2U_1-2U_2}{m}}$
$v = \sqrt{\frac{(2)\frac{(0.800~N~m^2)}{(0+0.200~m)}-(2)\frac{(0.800~N~m^2)}{(0.400~m+0.200~m)}}{0.500~kg}}$
$v = 3.27~m/s$
