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 $G$).

An object of mass $1 kg$ is taken to a height from the surface of earth which is equal to three times the radius of earth. The gain in potential energy of the object will be $….MJ$[If, $g =10\,ms ^{-2}$ and radius of earth $=6400\,km$ ]

Calculate energy needed for moving a mass of $4\,\,kg$ from the centre of the earth to its surface (in joule), if radius of the earth is $6400\,\, km$ and acceleration due to gravity at the surface of the earth is $g = 10 \,\,m/sec^2.$

Two stars of masses $3\times10^{31}\, kg$ each, and at distance $2\times10^{11}\, m$ rotate in a plane about their common centre of mass $O. A$ meteorite passes through $O$ moving perpendicular to the star’s rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at $O$ is : (Take Gravitational constant $G = 6.67\times10^{-11}\, Nm^2\, kg^{-2}$)

A satellite of mass $\mathrm{m}$ is launched vertically upwards with an initial speed $u$ from the surface of the earth. After it reaches height $\mathrm{R}$ ($R =$ radius of the earth), it ejects a rocket of mass $\frac{\mathrm{m}}{10}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is

($G$ is the gravitational constant: $\mathrm{M}$ is the mass of the earth)