If the escape velocity at the surface of a planet is $v_e$, and a projectile is launched vertically with a speed of $\frac{v_e}{\sqrt{2}}$, the maximum height reached by the projectile above the surface is:

If three particles each of mass $M$ are placed at the three corners of an equilateral triangle of side a, the forces exerted by this system on another particle of mass $M$ placed (i) at the mid point of a side and (ii) at the centre of the triangle are respectively

Assertion: If a particle projected horizontally just above the surface of earth with a speed greater than escape speed, then it will escape from gravitational influence of earth.Reason: Escape velocity is independent of its direction.

A spherical hollow is made in a lead sphere of radius $R$ such that its surface touches the outside surface of the lead sphere and passes through the centre. The mass of the lead sphere before hollowing was $M$. The force of attraction that this sphere would exert on a particle of mass m which lies at a distance $d\left( { > R} \right)$ from the centre of the lead sphere on the straight line joining the centres of the sphere and the hollow is

The mass of the moon is $\frac{1}{144}$ times the mass of a planet and its diameter is $\frac{1}{16}$ times the diameter of a planet. If the escape velocity on the planet is $v$, the escape velocity on the moon will be

Gravitational force between two point masses is found to be F when both are in vacuum. If both the masses are immersed in water, new gravitational force will be

A body weighed ${\rm{980N}}$ on the surface of the earth. its weight half away down to the center of the earth is [assuming earth to be of uniform density]

Two small and heavy spheres, each of mass M, are placed a distance $r$ apart on a horizontal surface. The gravitational potential at the mid-point on the line joining the centre of the spheres is

Escape velocity on earth is $11.2 \mathrm{~kms}^{-1}$, what would be the escape velocity on a planet whose mass is 1000 times and radius is 10 times that of earth?

Two particles of equal mass m go around a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle with respect to their center of mass is

A black hole has an escape velocity equal to the speed of light, c. If a planet of the same mass as the black hole had a radius R, what would be the radius of the black hole, assuming uniform density for both?