A body placed at a distance R₀ form the center of earth, starts moving from rest. The velocity of the body on reaching at the earth’s surface will be ( Rₑ = radius of earth and Mₑ = mass of earth)

The correct option is:

$\sqrt{2G{M}_e\left[\frac{1}{R_e}-\frac{1}{R_0}\right]}$

Step-by-Step Explanation:

To find the velocity of the body when it reaches the Earth's surface, we can use the law of conservation of mechanical energy. Since there are no non-conservative forces acting on the body (such as air resistance), the total mechanical energy of the body remains constant throughout its motion.

1. Initial State (at distance R₀ from the center of the Earth):
The body starts from rest, which means its initial velocity is zero. Thus, its initial kinetic energy is:

$K_i=0$

The gravitational potential energy of a body of mass m at a distance R₀ from the center of the Earth (mass M_e) is:

$U_i=-\frac{G{M}_{e}m}{R_0}$

Therefore, the initial total mechanical energy (E_i) is:

$E_i=K_i+U_i=-\frac{G{M}_{e}m}{R_0}$

2. Final State (on reaching the Earth's surface at distance R_e from the center):
Let the velocity of the body upon reaching the surface be v. The final kinetic energy is:

$K_f=\frac{1}{2}$

The final gravitational potential energy at the Earth's surface (distance R_e) is:

$U_f=-\frac{G{M}_{e}m}{R_e}$

Therefore, the final total mechanical energy (E_f) is:

$E_f=K_f+U_f=\frac{1}{2}$

3. Applying Conservation of Energy:
Equating the initial total mechanical energy to the final total mechanical energy:

$E_i=E_f$

$-\frac{G{M}_{e}m}{R_0}=\frac{1}{2}$

We can cancel out the mass of the body (m) from both sides of the equation:

$-\frac{G{M}_e}{R_0}=\frac{1}{2}$

Now, rearrange the terms to solve for $v^2$:

$\frac{1}{2}$

Factor out the common term $G{M}_e$ on the right-hand side:

Multiply both sides by 2:

$v^2=2G{M}_e\left[\frac{1}{R_e}-\frac{1}{R_0}\right]$

Taking the square root of both sides gives the final velocity of the body on reaching the Earth's surface:

$v=\sqrt{2G{M}_e\left[\frac{1}{R_e}-\frac{1}{R_0}\right]}$