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 solve the problem of finding the total mechanical energy of an object of mass \( m \) projected from the surface of the Earth with escape speed, we can follow these steps:

### Step 1: Understand Escape Speed
The escape speed is the minimum speed required for an object to break free from the gravitational pull of a planet without any additional propulsion. For Earth, the escape speed \( v_e \) is given by the formula:
\[ v_e = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity and \( R \) is the radius of the Earth.

### Step 2: Determine Kinetic Energy at Escape Speed
When the object is projected with escape speed, its kinetic energy \( KE \) at the surface of the Earth is calculated using the formula:
\[ KE = \frac{1}{2}mv_e^2 \]
Substituting the expression for escape speed:
\[ KE = \frac{1}{2}m(2gR) = mgR \]

### Step 3: Determine Potential Energy at the Surface of the Earth
The gravitational potential energy \( PE \) of the object at the surface of the Earth is given by:
\[ PE = -\frac{GMm}{R} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

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

### Step 5: Simplify the Expression
Using the relation \( g = \frac{GM}{R^2} \), we can express \( mgR \) as:
\[ mgR = m \left( \frac{GM}{R^2} \right) R = \frac{GMm}{R} \]
Thus, we can rewrite the total mechanical energy:
\[ E = \frac{GMm}{R} - \frac{GMm}{R} = 0 \]

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