A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The magnitude of the gravitational potential at a point sutuated at a/2 distance from the centre, will be :

A particle of mass m is thrown upwards from the surface of the earth, with a velocity u. The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth, is

A planet moving along an elliptical orbit is closest to the sun at a distance r₁ and farthest away at a distance of r₂. If $$v$$₁ and $$v$$₂ are the linear velocities at these points respectively, then the ratio $${{{v_1}} \over {{v_2}}}$$ is

The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius R₁ to another of radius R₂(R₂ > R₁) is