If mass of the earth is `M`, radius is `R`, and gravitational constant is `G`, then workdone to take `1kg` mass from earth surface to infinity will be

Options

A. `sqrt((GM)/(2R))`
B. `(GM)/R`
C. `sqrt((2GM)/R)`
D. `(GM)/(2R)`

Explanation

To find the work done to take a 1 kg mass from the surface of the Earth to infinity, we can use the concept of gravitational potential energy and escape velocity.

### Step-by-step Solution:

1. **Understand the Concept of Escape Velocity**:
The escape velocity (VE) is the minimum velocity required for an object to break free from the gravitational pull of a celestial body without any additional propulsion. For Earth, the escape velocity is given by the formula:
\[ V_E = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Calculate the Kinetic Energy at Escape Velocity**:
The kinetic energy (KE) of the mass (1 kg) when it reaches the escape velocity is given by:
\[ KE = \frac{1}{2} m V_E^2 \]
Substituting \( m = 1 \, \text{kg} \) and \( V_E \) from the previous step:
\[ KE = \frac{1}{2} \times 1 \times \left(\sqrt{\frac{2GM}{R}}\right)^2 = \frac{1}{2} \times 1 \times \frac{2GM}{R} = \frac{GM}{R} \]

3. **Work Done Against Gravitational Potential**:
The work done to take the mass from the surface of the Earth to infinity is equal to the kinetic energy required to escape the gravitational field of the Earth. Thus, the work done (W) is:
\[ W = KE = \frac{GM}{R} \]

4. **Final Result**:
Therefore, the work done to take a 1 kg mass from the surface of the Earth to infinity is:
\[ W = \frac{GM}{R} \]