A body attains a height equal to the radius of the earth when projected from earth's surface the velocity of body with which it was projected is

Options

A. `sqrt((GM)/(R))`
B. `sqrt((2GM)/(R))`
C. `sqrt((5)/(4)(GM)/(R))`
D. `sqrt((3GM)/(R))`

Answer

To solve the problem, we need to find the velocity with which a body must be projected from the Earth's surface to reach a height equal to the radius of the Earth (R). We will use the principle of conservation of energy.

### Step-by-Step Solution:

1. **Identify Initial and Final States**:
- The initial state is when the body is projected from the Earth's surface.
- The final state is when the body reaches a height equal to the radius of the Earth (R).

2. **Write the Conservation of Energy Equation**:
The total mechanical energy at the initial position (when the body is projected) must equal the total mechanical energy at the final position (at height R).
\[ K_{\text{initial}} + U_{\text{initial}} = K_{\text{final}} + U_{\text{final}} \]

3. **Define Kinetic and Potential Energy**:
- Initial Kinetic Energy (K_initial):
\[ K_{\text{initial}} = \frac{1}{2} m v^2 \]
- Initial Potential Energy (U_initial):
\[ U_{\text{initial}} = -\frac{GMm}{R} \]
- Final Kinetic Energy (K_final):
\[ K_{\text{final}} = 0 \quad (\text{at maximum height, velocity is zero}) \]
- Final Potential Energy (U_final):
\[ U_{\text{final}} = -\frac{GMm}{2R} \quad (\text{at height R, distance from center is } 2R) \]

4. **Set Up the Equation**:
Substituting the expressions for kinetic and potential energy into the conservation of energy equation:
\[ \frac{1}{2} mv^2 - \frac{GMm}{R} = 0 - \frac{GMm}{2R} \]

5. **Simplify the Equation**:
Rearranging gives:
\[ \frac{1}{2} mv^2 = \frac{GMm}{R} - \frac{GMm}{2R} \]
\[ \frac{1}{2} mv^2 = \frac{GMm}{2R} \]

6. **Cancel Out Mass (m)**:
Since mass (m) appears on both sides, we can cancel it:
\[ \frac{1}{2} v^2 = \frac{GM}{2R} \]

7. **Solve for Velocity (v)**:
Multiply both sides by 2:
\[ v^2 = \frac{GM}{R} \]
Taking the square root gives:
\[ v = \sqrt{\frac{GM}{R}} \]

### Final Answer:
The velocity with which the body must be projected from the Earth's surface is:
\[ v = \sqrt{\frac{GM}{R}} \]