A particle is kept at rest at a distance R (earth's radius) above the earth's surface. The minimum speed with which it should be projected so that is does not return is

Options

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

Answer

To solve the problem of finding the minimum speed with which a particle should be projected from a height equal to the Earth's radius (R) so that it does not return, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem:**
- The particle is at a distance R above the Earth's surface. Therefore, the total distance from the center of the Earth (denoted as O) to the particle (P) is:
\[ OP = R + R = 2R \]

2. **Potential Energy at the Height:**
- The gravitational potential energy (U) of the particle at this height (2R) is given by:
\[ U = -\frac{GMm}{r} \]
- Here, \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the particle, and \( r \) is the distance from the center of the Earth. For our case, \( r = 2R \):
\[ U = -\frac{GMm}{2R} \]

3. **Potential Energy at Infinity:**
- The potential energy at an infinite distance from the Earth is:
\[ U_{\infty} = 0 \]

4. **Change in Potential Energy:**
- The change in potential energy (ΔU) when the particle moves from height 2R to infinity is:
\[ \Delta U = U_{\infty} - U = 0 - \left(-\frac{GMm}{2R}\right) = \frac{GMm}{2R} \]

5. **Kinetic Energy:**
- The kinetic energy (K.E) of the particle when it is projected with speed \( v \) is given by:
\[ K.E = \frac{1}{2} mv^2 \]

6. **Applying the Energy Conservation Principle:**
- For the particle to not return, the kinetic energy must equal the change in potential energy:
\[ \frac{1}{2} mv^2 = \frac{GMm}{2R} \]

7. **Solving for the Speed \( v \):**
- Cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{1}{2} v^2 = \frac{GM}{2R} \]
- Multiply both sides by 2:
\[ v^2 = \frac{GM}{R} \]
- Taking the square root gives:
\[ v = \sqrt{\frac{GM}{R}} \]

8. **Final Expression:**
- The minimum speed with which the particle should be projected so that it does not return is:
\[ v = \sqrt{\frac{GM}{R}} \]