The distance between two moving particles `P` and `Q` at any time is a.If `v_(r)` be their relative velocity and if `u` and `v` be the components of `v_(r)`, along and perpendicular to `PQ`.The closest distance between `P` and `Q` and time that elapses before they arrive at their nearest distance is

Options

A. `(av_(1))/(v^(2))`
B. `(av_(2))/(v^(2))`
C. `(av)/(v_(1)^(2))`
D. `(av)/(v_(2)^(2))`

Answer

To solve the problem, we need to find the closest distance between two moving particles \( P \) and \( Q \) and the time that elapses before they arrive at their nearest distance.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the distance between the two particles \( P \) and \( Q \) at any time be \( A \).
- The relative velocity \( v_r \) of the two particles can be resolved into two components:
- \( u \): the component of \( v_r \) along the line joining \( P \) and \( Q \) (PQ).
- \( v \): the component of \( v_r \) perpendicular to PQ.

2. **Finding the Closest Distance**:
- The closest distance between the two particles occurs when the line connecting them is perpendicular to the relative velocity vector.
- This closest distance can be represented as \( QR \), where \( QR \) is the perpendicular distance from the line of motion to the point of closest approach.
- Using trigonometry, we can express \( QR \) in terms of \( A \) and the angle \( \alpha \) between the relative velocity and the line joining \( P \) and \( Q \):
\[ QR = A \cos(\alpha) \]

3. **Finding the Time to Closest Approach**:
- The time \( T \) taken to reach the closest distance is given by the formula:
\[ T = \frac{\text{Distance}}{\text{Velocity}} \]
- Here, the distance is \( QR \) and the velocity is the component of the relative velocity perpendicular to PQ, which is \( v \).
- Therefore, we can write:
\[ T = \frac{QR}{v} = \frac{A \cos(\alpha)}{v} \]

4. **Relating \( \cos(\alpha) \) to the Relative Velocity**:
- From the triangle formed by the components of the relative velocity, we can express \( \cos(\alpha) \) as:
\[ \cos(\alpha) = \frac{v}{v_r} \]
- Substituting this back into the time equation gives:
\[ T = \frac{A \cdot \frac{v}{v_r}}{v} = \frac{A}{v_r} \]

5. **Final Expression**:
- The closest distance \( QR \) can be expressed as:
\[ QR = A \cdot \frac{v}{v_r} \]
- The time \( T \) taken to reach this closest distance is:
\[ T = \frac{A}{v_r} \]

### Conclusion:
- The closest distance between particles \( P \) and \( Q \) is \( A \cdot \frac{v}{v_r} \).
- The time elapsed before they arrive at their nearest distance is \( \frac{A}{v_r} \).