What is the magnitude of potential energy at infinity distance ?

A small asteroid is orbiting around the sun in a circular orbit of radius $r_0$ with speed $v_0$. A rocket is launched from the asteroid with speed $v=\alpha v_0$, where $v$ is the speed relative to the sun. The highest value of $\alpha$ for which the rocket will remain bound to the solar system is (ignoring gravity due to the asteroid and eff ects of other planets)

The particles $A$ and $B$ of mass $m$ each are separated by a distance $r$. Another particle $C$ of mass $M$ is placed at the midpoint of $A$ and $B$. Find the work done in taking $C$ to a point equidistant $r$ from $A$ and $B$ without acceleration $(G=$ Gravitational constant and only gravitational interaction between $A, B$ and $C$ is considered) 

A particle of mass $m$ moves around the origin in a potential $\frac{1}{2} m \omega^{2} r^{2}$, where $r$ is the distance from the origin. Applying the Bohr's model in this case, the radius of the particle in its $n$th orbit in terms of $a=\sqrt{h /(2 \pi m \omega)}$ is

A satellite of mass $m$ is placed at a distance $r$ from the centre of earth (mass $M$). The mechanical energy of the satellite is