The distance between two particles is decreasing at the rate of 6 m/sec,when they travel in opposite direction to each other. If these particles travel with same speeds and in the same direction, then the separation increase at the rate of 4 m/sec. The particles have speed as

Options

A. 5 m/sec, 1 m/sec
B. 4 m/sec, 1 m/sec
C. 4 m/sec, 2 m/sec
D. 5 m/sec, 2 m/sec

Explanation

To solve the problem, we need to analyze the motion of two particles under two different conditions: when they are moving towards each other and when they are moving in the same direction.

### Step-by-Step Solution:

1. **Define Variables**:
Let the speed of each particle be \( v \). Since both particles have the same speed, we can denote their speeds as \( v_1 = v \) and \( v_2 = v \).

2. **Condition 1: Particles Moving Towards Each Other**:
When the two particles are moving towards each other, their relative speed is the sum of their speeds. According to the problem, the distance between them is decreasing at a rate of 6 m/s. Therefore, we can write:
\[ v_1 + v_2 = 6 \text{ m/s} \]
Substituting \( v_1 \) and \( v_2 \):
\[ v + v = 6 \implies 2v = 6 \implies v = 3 \text{ m/s} \]

3. **Condition 2: Particles Moving in the Same Direction**:
When the two particles are moving in the same direction, their relative speed is the difference of their speeds. The problem states that the separation is increasing at a rate of 4 m/s. Therefore, we can write:
\[ v_1 - v_2 = 4 \text{ m/s} \]
Since both particles have the same speed, this equation simplifies to:
\[ v - v = 4 \implies 0 = 4 \]
This indicates that we need to reconsider the interpretation of the speeds. Since they are moving in the same direction, we should consider that one particle is faster than the other.

4. **Revising Condition 2**:
Let's denote the speed of one particle as \( v \) and the other as \( v + x \) (where \( x \) is the difference in speed). Then:
\[ (v + x) - v = 4 \implies x = 4 \text{ m/s} \]
Now we have:
\[ v_1 = v \quad \text{and} \quad v_2 = v + 4 \]

5. **Substituting Back into Condition 1**:
Now we can substitute back into the first condition:
\[ v + (v + 4) = 6 \implies 2v + 4 = 6 \implies 2v = 2 \implies v = 1 \text{ m/s} \]

6. **Finding the Speeds**:
Now we can find the speeds of both particles:
\[ v_1 = 1 \text{ m/s} \quad \text{and} \quad v_2 = 1 + 4 = 5 \text{ m/s} \]

### Conclusion:
The speeds of the two particles are \( 1 \text{ m/s} \) and \( 5 \text{ m/s} \).