The radius of the earth is $R$. If a body is taken to a height $3R$ from the surface of the earth,then the change in potential energy will be:

The work done to raise a mass $m$ from the surface of the earth to a height $h$,which is equal to the radius of the earth,is

$A$ body is thrown from the surface of the earth with velocity $u \ m \ s^{-1}$. The maximum height in meters above the surface of the earth up to which it will reach is ($R=$ radius of the earth,$g=$ acceleration due to gravity).

If the gravitational acceleration at the surface of the Earth is $g$,then the increase in potential energy in lifting an object of mass $m$ to a height equal to half of the radius of the Earth from the surface will be:

An asteroid is moving directly towards the centre of the Earth. When at a distance of $10 R$ ($R$ is the radius of the Earth) from the Earth's centre,it has a speed of $12 \; km/s$. Neglecting the effect of Earth's atmosphere,what will be the speed of the asteroid when it hits the surface of the Earth? (Escape velocity from the Earth is $11.2 \; km/s$). Give your answer to the nearest integer in $km/s$.

$A$ 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 $R = 6400 \ km$):