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 it 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 must be given to a body of mass \( m \) situated at a distance \( 4R_e \) above the Earth's surface in order to escape Earth's gravitational pull, we can follow these steps:

### Step 1: Understand the situation
The body is located at a distance of \( 4R_e \) above the Earth's surface. The total distance from the center of the Earth is therefore:
\[ d = R_e + 4R_e = 5R_e \]

### Step 2: Use the concept of gravitational potential energy
The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{G M_e m}{r} \]
where \( G \) is the gravitational constant, \( M_e \) is the mass of the Earth, and \( m \) is the mass of the body.

### Step 3: Calculate the initial potential energy
At a distance of \( 5R_e \) from the center of the Earth, the initial potential energy \( U_i \) is:
\[ U_i = -\frac{G M_e m}{5R_e} \]

### Step 4: Determine the final potential energy
When the body escapes to infinity, the potential energy \( U_f \) approaches zero:
\[ U_f = 0 \]

### Step 5: Apply the conservation of mechanical energy
The conservation of mechanical energy states:
\[ K_i + U_i = K_f + U_f \]
where \( K_i \) is the initial kinetic energy and \( K_f \) is the final kinetic energy. For minimum energy required to escape, we assume that the final kinetic energy \( K_f \) is zero (the body just reaches infinity with no velocity):
\[ K_i + U_i = 0 \]

### Step 6: Solve for initial kinetic energy
Rearranging the equation gives:
\[ K_i = -U_i \]
Substituting the expression for \( U_i \):
\[ K_i = \frac{G M_e m}{5R_e} \]

### Step 7: Relate gravitational constant to surface gravity
We know that the gravitational acceleration at the surface of the Earth \( g \) is given by:
\[ g = \frac{G M_e}{R_e^2} \]
Thus, we can express \( G M_e \) as:
\[ G M_e = g R_e^2 \]
Substituting this back into our expression for \( K_i \):
\[ K_i = \frac{g R_e^2 m}{5R_e} = \frac{g m R_e}{5} \]

### Conclusion
The minimum energy \( E \) that must be given to the body to escape the Earth's gravitational field is:
\[ E = \frac{g m R_e}{5} \]