Question

A planet of mass M and radius R has no atmosphere. The escape velocity at its surface is ve. An object of mass m is at rest a distance r from the center of the planet, where r>>R. The particle falls to the surface of the planet. The total mechanical energy of the particle at the surface of the planet is closest to

Answer 1

Answer:

Explanation:

Expression for escape velocity

ve = $\sqrt{\frac{2GM}{R} }$

ve² R / 2 = GM

M is mass of the planet , R is radius of the planet .

At distance r >> R , potential energy of object

= $\frac{-GMm}{r}$

Since the object is at rest at that point , kinetic energy  will be zero .

Total mechanical energy  = $\frac{-GMm}{r}$ + 0 = $\frac{-GMm}{r}$

Putting the value of GM = ve² R / 2

Total mechanical energy  = ve² Rm / 2 r

This mechanical energy will be conserved while falling down on the earth due to law of conservation of mechanical energy  . So at surface of the earth , total mechanical energy

=  ve² Rm / 2 r