An object is allowed to fall from a height $R$ above the earth,where $R$ is the radius of the earth. Its velocity when it strikes the earth's surface,ignoring air resistance,will be:

Two uniform solid spheres of equal radii $R$,but mass $M$ and $4M$ have a centre-to-centre separation $6R$,as shown in the figure. The two spheres are held fixed. $A$ projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the centre of the second sphere. Obtain an expression for the minimum speed $v$ of the projectile so that it reaches the surface of the second sphere.

$A$ body of mass $m$ is placed on the earth's surface. It is taken from the earth's surface to a height $h = 3R$. The change in gravitational potential energy of the body is

Suppose two planets (spherical in shape) of radii $R$ and $2R$,but mass $M$ and $9M$ respectively have a centre-to-centre separation $8R$ as shown in the figure. $A$ satellite of mass $m$ is projected from the surface of the planet of mass $M$ directly towards the centre of the second planet. The minimum speed $v$ required for the satellite to reach the surface of the second planet is $\sqrt{\frac{a}{7} \frac{GM}{R}}$,then the value of $a$ is $....$
[Given: The two planets are fixed in their position]

The escape velocity from a planet is $V_e$. $A$ tunnel is dug along the diameter of the planet and a small body is dropped into it. The speed of the body at the centre of the planet will be

$A$ rocket of mass $M$ is launched vertically from the surface of the earth with an initial speed $V$. Assuming the radius of the earth to be $R$ and negligible air resistance,the maximum height attained by the rocket above the surface of the earth is