When escape velocity is given to a particle on surface of earth, its total energy is

Options

A. zero
B. greater than zero
C. less than zero
D. `-GMm//2R`

Explanation

To solve the question regarding the total energy of a particle when escape velocity is given to it on the surface of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Escape Velocity**:
The escape velocity (\(v_e\)) from the surface of the Earth is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \(G\) is the universal gravitational constant, \(M\) is the mass of the Earth, and \(R\) is the radius of the Earth.

2. **Calculate Kinetic Energy (KE)**:
The kinetic energy of the particle when it is given escape velocity is:
\[ KE = \frac{1}{2} mv_e^2 \]
Substituting the expression for escape velocity:
\[ KE = \frac{1}{2} m \left(\sqrt{\frac{2GM}{R}}\right)^2 = \frac{1}{2} m \cdot \frac{2GM}{R} = \frac{mGM}{R} \]

3. **Calculate Potential Energy (PE)**:
The gravitational potential energy of the particle at the surface of the Earth is given by:
\[ PE = -\frac{GMm}{R} \]

4. **Total Energy (E)**:
The total energy of the particle is the sum of its kinetic energy and potential energy:
\[ E = KE + PE \]
Substituting the values we calculated:
\[ E = \frac{mGM}{R} - \frac{GMm}{R} \]
Simplifying this:
\[ E = \frac{mGM}{R} - \frac{mGM}{R} = 0 \]

5. **Conclusion**:
Therefore, when escape velocity is given to a particle on the surface of the Earth, its total energy is:
\[ E = 0 \]