A rock of mass \(m\) is thrown radially outward from the surface of a spherical, airless moon of radius \(R\). From Newton's second law its acceleration is \(\ddot{r}=-G M / r^{2}\), where \(M\) is the moon's mass and \(r\) is the distance from the moon's center to the rock. The energy of the rock is conserved, so \((1 / 2) m \dot{r}^{2}-G M m / r=E=\) constant.

(a) Show by differentiating this equation that energy conservation is a first integral of \(F=m \ddot{r}\) in this case.

(b) What is the minimum value of \(E\), in terms of given parameters, for which the rock will escape from the moon?

(c) For this case what is \(\dot{r}(t)\), the velocity of the rock as a function of time since it was thrown?