Satellites and Binding Energy EX 3

A probe will escape the gravitational attraction of the Sun and travel into an interstellar space when the system Sun - interstellar probe becomes unbound or when the total energy $E$ of this system becomes

$E\geqslant 0$

To solve this problem we have to first determine how much energy this system has and then we will be able to determine how much more it needs to become unbound. We know that the probe orbits the Sun at a distance $r=1.5\times 10^{11}m$. At this distance the energy of the system is

$E={\frac {1}{2}}U=-{\frac {1}{2}}{GMm \over r}=-{\frac {1}{2}}{(6.67\times 10^{-11})(2\times 10^{30})(10) \over 1.5\times 10^{11}}=-4.45\times 10^{9}J$

This is the initial energy of the probe before the rocket was fired. The final energy should be larger or equal to zero and therefore, the minimum amount of energy that the rocket must supply, is $4.45\times 10^{9}J$.

Note that the rocket must do more work than $4.45\times 10^{9}J$. The rocket itself has a large mass of fuel and some fuel will increase its energy before it is used up.