Orbital velocity of a satellite in its orbit (around earth) of radius r is v. It collides with another body in its orbit and comes to rest just after the collision. Taking the radius of earth as R, the speed with which it will fall on the surface of earth will be :

Options

A. `vsqrt(((r)/(R)-1))`
B. `vsqrt(2((r)/(R)-1))`
C. `(v)/(sqrt(2((r)/(R)+1))`
D. `vsqrt(2((r)/(R)+1))`

Explanation

To solve the problem, we need to determine the speed with which a satellite will fall to the surface of the Earth after colliding with another body and coming to rest in its orbit. We will use the principles of energy conservation and the formulas related to gravitational potential energy and kinetic energy.

### Step-by-Step Solution:

1. **Understanding Orbital Velocity**:
The orbital velocity \( v \) of a satellite in a circular orbit of radius \( r \) around the Earth is given by the formula:
\[
v = \sqrt{\frac{GM}{r}}
...