The total mechanical energy of an object of mass m projected from surface of earth with escape speed is

Options

A. Zero
B. Infinite
C. `-(GMm)/(2R)`
D. `-(GMm)/(3R)`

Answer

To find the total mechanical energy of an object of mass \( m \) projected from the surface of the Earth with escape speed, we will follow these steps:

### Step 1: Understand Escape Speed
Escape speed is the minimum speed required for an object to break free from the gravitational attraction of a celestial body without any additional propulsion. For Earth, the escape speed \( v \) is given by the formula:
\[ v = \sqrt{\frac{2GM}{R}} \]
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
The kinetic energy (KE) of the object when it is projected with escape speed is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting the escape speed into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\sqrt{\frac{2GM}{R}}\right)^2 = \frac{1}{2} m \cdot \frac{2GM}{R} = \frac{mGM}{R} \]

### Step 3: Calculate Gravitational Potential Energy
The gravitational potential energy (PE) of the object at the surface of the Earth is given by:
\[ PE = -\frac{GMm}{R} \]
This negative sign indicates that the gravitational force is attractive.

### Step 4: Calculate Total Mechanical Energy
The total mechanical energy (E) is the sum of kinetic energy and potential energy:
\[ E = KE + PE \]
Substituting the expressions for kinetic and potential energy:
\[ E = \frac{mGM}{R} - \frac{GMm}{R} \]
This simplifies to:
\[ E = \frac{mGM}{R} - \frac{mGM}{R} = 0 \]

### Conclusion
Thus, the total mechanical energy of an object of mass \( m \) projected from the surface of the Earth with escape speed is:
\[ \boxed{0} \]