A body of mass m is situated at a distance `4R_(e)` above the earth's surface, where `R_(e)` is the radius of earth. How much minimum energy be given to the body so that may escape

Options

A. `mgR_(e)`
B. `2mgR_(e)`
C. `(mgR_(e))/5`
D. `(mgR_(e))/16`

Explanation

To solve the problem of how much minimum energy is required for a body of mass \( m \) situated at a distance of \( 4R_e \) above the Earth's surface to escape the Earth's gravitational field, we can follow these steps:

### Step 1: Determine the distance from the center of the Earth
The distance \( d \) from the center of the Earth to the body is given by:
\[ d = R_e + 4R_e = 5R_e \]

### Step 2: Write the formula for gravitational potential energy
The gravitational potential energy \( U \) of the body at a distance \( d \) from the center of the Earth is given by:
\[ U = -\frac{G M m}{d} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the body,
- \( d \) is the distance from the center of the Earth.

### Step 3: Substitute the distance into the potential energy formula
Substituting \( d = 5R_e \) into the potential energy formula gives:
\[ U = -\frac{G M m}{5R_e} \]

### Step 4: Determine the energy required to escape
To escape the gravitational field of the Earth, the total mechanical energy must be zero. Therefore, we need to provide enough energy \( E \) to make the total energy (potential energy + provided energy) equal to zero:
\[ E + U = 0 \]
Thus, we have:
\[ E = -U = \frac{G M m}{5R_e} \]

### Step 5: Express \( G M \) in terms of \( g \)
We know that the acceleration due to gravity \( g \) at the Earth's surface is given by:
\[ g = \frac{G M}{R_e^2} \]
From this, we can express \( G M \) as:
\[ G M = g R_e^2 \]

### Step 6: Substitute \( G M \) into the energy equation
Substituting \( G M \) into the energy equation gives:
\[ E = \frac{g R_e^2 m}{5R_e} = \frac{g m R_e}{5} \]

### Final Answer
Thus, the minimum energy required to be given to the body so that it may escape the Earth's gravitational field is:
\[ E = \frac{g m R_e}{5} \]

---