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) 

Two identical satellites $A$ and $B$ are circulating round the earth at the height of $R$ and $2R$ respectively, (where $R$ is radius of the earth). The ratio of kinetic energy of $A$ to that of $B$ is

For a satellite if the time of revolution is $T$, then its $K.E.$ is proportional to

Two stars each of one solar mass $\left(=2 \times 10^{30}\; kg \right)$ are approaching each other for a head on collision. When they are a distance $10^{9}\; km$, their speeds are negligible. What is the speed with which they collide? The radius of each star is $10^{4}\; km$. Assume the stars to remain undistorted until they collide. (Use the known value of $G$).

A satellite of mass $m,$ initially at rest on the earth, is launched into a circular orbit at a height equal to the radius of the earth. The minimum energy required is