The distance between two moving particles at any time is a. If `v` be their relative velocity and `v_(1)` and `v_(2)` be the components of `v` along and perpendicular to a. The time when they are closest to each other is

Options

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

Answer

To solve the problem step by step, we will analyze the situation involving two moving particles and their relative motion.

### Step 1: Understand the Problem
We have two particles moving such that the distance between them at any time is \( A \). The relative velocity \( v \) is given, and we need to find the time when they are closest to each other. The components of the relative velocity are \( v_1 \) (along the line connecting the two particles) and \( v_2 \) (perpendicular to that line).

**Hint:** Visualize the situation by drawing a diagram of the two particles and the vectors representing their velocities.

### Step 2: Identify the Components of Relative Velocity
The relative velocity \( v \) can be broken down into two components:
- \( v_1 \): the component of the relative velocity along the line connecting the two particles.
- \( v_2 \): the component of the relative velocity perpendicular to that line.

**Hint:** Remember that the relative velocity can be represented as a vector triangle where \( v \) is the hypotenuse, and \( v_1 \) and \( v_2 \) are the two legs.

### Step 3: Determine the Approaching Velocity
The component \( v_1 \) is the effective velocity that brings the two particles closer together. The time \( T \) when they are closest is determined by how long it takes for the distance \( A \) to be covered at this approaching velocity \( v_1 \).

**Hint:** The time to cover a distance can be calculated using the formula \( T = \frac{\text{Distance}}{\text{Velocity}} \).

### Step 4: Write the Expression for Time
Using the distance \( A \) and the approaching velocity \( v_1 \), we can express the time \( T \) when the particles are closest:
\[ T = \frac{A}{v_1} \]

**Hint:** Substitute \( v_1 \) in terms of \( v \) and \( v_2 \) if necessary.

### Step 5: Relate \( v_1 \) to \( v \) and \( v_2 \)
From the triangle formed by the velocities, we can relate \( v_1 \) and \( v_2 \) using trigonometric identities:
\[ v_1 = v \cos(\theta) \]
where \( \theta \) is the angle between \( v \) and \( v_1 \).

**Hint:** Use the relationship between the components of the velocity to express \( v_1 \) in terms of \( v \) and \( v_2 \).

### Step 6: Substitute \( v_1 \) into the Time Equation
Substituting \( v_1 \) into the time equation gives:
\[ T = \frac{A}{v \cos(\theta)} \]

**Hint:** Recognize that \( \cos(\theta) \) can be expressed as \( \frac{v_1}{v} \).

### Step 7: Final Expression for Time
After substituting and simplifying, we find:
\[ T = \frac{A v_1}{v^2} \]

**Hint:** Ensure that all terms are correctly placed and that the units are consistent.

### Conclusion
The time when the two particles are closest to each other is given by:
\[ T = \frac{A v_1}{v_2^2} \]
Thus, the correct option is option 1.