A particle on earth's surface is given a velocity euqal to its escape velocity. Its total mechanical energy with zero potential energy reference at infinite separation will be:

Options

A. negative
B. positive
C. zero
D. infinite

Answer

To determine the total mechanical energy of a particle on Earth's surface when it is given a velocity equal to its escape velocity, we can follow these steps:

### Step 1: Understand Escape Velocity
Escape velocity is the minimum velocity needed for an object to break free from the gravitational attraction of a celestial body, without any further propulsion. For Earth, the escape velocity (v_e) is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 2: Calculate Kinetic Energy at Earth's Surface
The kinetic energy (KE) of the particle when it is given the escape velocity is calculated using the formula:
\[ KE = \frac{1}{2}mv^2 \]
Substituting \( v \) with the escape velocity \( v_e \):
\[ KE = \frac{1}{2}m(v_e)^2 = \frac{1}{2}m\left(\frac{2GM}{R}\right) = \frac{GMm}{R} \]

### Step 3: Calculate Gravitational Potential Energy at Earth's Surface
The gravitational potential energy (PE) of the particle at the surface of the Earth is given by:
\[ PE = -\frac{GMm}{R} \]
This value is negative because gravitational potential energy is defined to be zero at an infinite distance from the mass.

### Step 4: Calculate Total Mechanical Energy
The total mechanical energy (E) of the particle is the sum of its kinetic energy and potential energy:
\[ E = KE + PE \]
Substituting the values we calculated:
\[ E = \frac{GMm}{R} - \frac{GMm}{R} = 0 \]

### Conclusion
Thus, the total mechanical energy of the particle when it is given a velocity equal to its escape velocity is:
\[ E = 0 \]

### Final Answer
The total mechanical energy with zero potential energy reference at infinite separation will be **0**.

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