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 body of mass $m$ kg. starts falling from a point $2R$ above the Earth’s surface. Its kinetic energy when it has fallen to a point $‘R’$ above the Earth’s surface [$R-$Radius of Earth, $M-$Mass of Earth, $G-$Gravitational Constant]

A satellite is revolving around a planet of mass $'M'$ in an elliptical orbit of semi-major axis $'a'$. Then the speed of satellite when it is at a distance $(a/2)$ from the planet

A satellite of mass $m$ is in circular orbit of radius $3 R_E$ about earth (mass of earth $M_{E}$, radius of earth $R_{E}$). How much additional energy is required to transfer the satellite to an orbit of radius $9 R_{E}$?

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