In above question 1, find the speed of each speed of each particle, when the separation reduces to half its initial value

Options

A. `sqrt((Gm)/(d))`
B. `sqrt((2Gm)/(d))`
C. `sqrt((Gm)/(2d))`
D. None of these

Explanation

To solve the problem of finding the speed of each particle when the separation reduces to half its initial value, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Energy**:
The gravitational potential energy (E_initial) between two particles of mass \( m \) separated by a distance \( d \) is given by:
\[ E_i = -\frac{G m m}{d} \]
where \( G \) is the gravitational constant.

2. **Final Separation**:
When the separation reduces to half its initial value, the new distance becomes \( \frac{d}{2} \). The gravitational potential energy (E_final) at this new separation is:
\[ E_f = -\frac{G m m}{\frac{d}{2}} = -\frac{2G m m}{d} \]

3. **Applying the Conservation of Energy**:
The total mechanical energy is conserved. Therefore, the initial energy plus the kinetic energy of the particles must equal the final energy:
\[ E_i + K_i = E_f + K_f \]
Assuming both particles start from rest, the initial kinetic energy \( K_i = 0 \).

4. **Setting Up the Equation**:
The final kinetic energy when the separation is half can be expressed as:
\[ K_f = \frac{1}{2} mv^2 + \frac{1}{2} mv'^2 \]
Since both particles are identical, we can denote their speeds as \( v \) and \( v' \) (which will be equal). Thus:
\[ K_f = mv^2 \]

5. **Equating Energies**:
Now we can set up the equation:
\[ -\frac{G m m}{d} + 0 = -\frac{2G m m}{d} + mv^2 \]
Rearranging gives:
\[ mv^2 = -\frac{G m m}{d} + \frac{2G m m}{d} \]
Simplifying:
\[ mv^2 = \frac{G m m}{d} \]

6. **Solving for Speed**:
Dividing both sides by \( m \):
\[ v^2 = \frac{G m}{d} \]
Taking the square root:
\[ v = \sqrt{\frac{G m}{d}} \]

### Final Answer:
The speed of each particle when the separation reduces to half its initial value is:
\[ v = \sqrt{\frac{G m}{d}} \]