Two bodies of masses $m_1$ and $m_2$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

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) 

Four particles $A, B, C$ and $D$ each of mass $m$ are kept at the corners of a square of side $L$. Now the particle $D$ is taken to infinity by an external agent keeping the other particles fixed at their respective positions. The work done by the gravitational force acting on the particle $D$ during its movement is ……….

The mass of a spaceship is $1000\ kg$. It is to be launched from the earth's surface out into free space. The value of $g$ and $R$( radius of earth) are $10\ m/s^2 $ and $6400\ km$ respectively. The required energy for this work will be

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)