The change in the gravitational potential energy when a body of mass $m$ is raised to a height $h = nR$ above the surface of the Earth is (where $R$ is the radius of the Earth).

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]

$A$ body of mass $m$ is taken from the earth's surface to a height equal to twice the radius $(R)$ of the earth. The change in potential energy of the body will be

$A$ body of mass $m$ starts falling from a distance $3R$ above the Earth's surface. When it reaches a distance $R$ above the surface of the Earth (radius $R$,mass $M$),its kinetic energy is:

$A$ projectile is thrown straight upward from the Earth's surface with an initial speed $v = \alpha v_E$,where $\alpha$ is a constant and $v_E$ is the escape speed. The projectile travels up to a height $h = 800 \ km$ from the Earth's surface before it comes to rest. The value of the constant $\alpha$ is (Radius of the Earth $R = 6400 \ km$):

$A$ body is released from a height equal to the radius $(R)$ of the earth. The velocity of the body when it strikes the surface of the earth will be: (Given $g$ = acceleration due to gravity on the earth.)