A small spacecraft of mass m is initially at rest on the surface of an airless planet of mass M and radius R. Neglect atmospheric effects and rotation of the planet. The spacecraft’s engines fire long enough to give it just enough speed to escape permanently from the planet’s gravitational influence (escape speed).
(1) Using the conventional reference where gravitational potential energy is zero at infinite separation, derive an expression for the minimum launch speed vesc required for escape.
(2) An engineer suggests using an alternate reference in which the gravitational potential energy is defined to be zero at the planet’s surface. With this reference choice, write expressions for the initial and final total mechanical energy of the spacecraft–planet system when the spacecraft is launched at speed vesc and ultimately reaches an infinite distance with zero speed.
(3) Using your expressions from part (b), show algebraically that the value of vesc found in part (a) is unchanged by the alternate reference choice, even though the individual potential–energy terms differ.
(4) Explain why defining gravitational potential energy to be zero at the planet’s surface might be convenient for some engineering calculations, yet makes no difference to any physical predictions about the spacecraft’s motion.