Binding energy of a particle on the surface of earth is E. Kinetic energy grater than E is given to this particle. Then total energy of particle will become

Options

A. zero
B. infinite
C. `gt0`
D. `lt0`

Answer

To solve the problem, we need to find the total energy of a particle on the surface of the Earth when it has been given kinetic energy greater than its binding energy \( E \). Here’s a step-by-step solution:

### Step 1: Understand Binding Energy
The binding energy \( E \) of a particle on the surface of the Earth is defined as the energy required to remove the particle from the gravitational influence of the Earth. Mathematically, it can be expressed as:
\[ E = -\frac{GMm}{R} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the particle,
- \( R \) is the radius of the Earth.

### Step 2: Define Kinetic Energy
The problem states that kinetic energy greater than \( E \) is given to the particle. Let’s denote this kinetic energy as \( K \). Since \( K > E \), we can express \( K \) as:
\[ K = E + \Delta E \]
where \( \Delta E \) is some positive energy.

### Step 3: Calculate Total Energy
The total energy \( T \) of the particle is the sum of its kinetic energy \( K \) and its potential energy \( U \). The potential energy \( U \) of the particle at the surface of the Earth is given by:
\[ U = -\frac{GMm}{R} \]
Thus, the total energy \( T \) can be expressed as:
\[ T = K + U \]
Substituting the expressions for \( K \) and \( U \):
\[ T = (E + \Delta E) + \left(-\frac{GMm}{R}\right) \]
Since \( E = -\frac{GMm}{R} \), we can substitute \( E \) into the equation:
\[ T = \left(-\frac{GMm}{R} + \Delta E\right) + \left(-\frac{GMm}{R}\right) \]
This simplifies to:
\[ T = -\frac{GMm}{R} - \frac{GMm}{R} + \Delta E \]
\[ T = -\frac{2GMm}{R} + \Delta E \]

### Step 4: Analyze Total Energy
Since \( \Delta E \) is positive and \( -\frac{2GMm}{R} \) is negative, the total energy \( T \) could be greater than zero if \( \Delta E \) is sufficiently large. However, we need to determine the overall sign of \( T \).

### Conclusion
The total energy \( T \) can be expressed as:
\[ T = \Delta E - \frac{2GMm}{R} \]
If \( \Delta E \) is greater than \( \frac{2GMm}{R} \), then \( T \) will be positive. If not, it will remain negative.

Thus, the final answer is that the total energy of the particle will depend on the value of \( \Delta E \) relative to \( \frac{2GMm}{R} \).