The distance between two particles is decreasing at the rate of 6 m/sec. 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 information given about the two particles and their speeds. Let's break it down step by step.

### Step 1: Understand the problem
We have two particles moving towards each other, and the distance between them is decreasing at a rate of 6 m/s. When they are moving in the same direction, the distance between them increases at a rate of 4 m/s.

### Step 2: Set up the equations
1. **When the particles are moving towards each other:**
- Let the speeds of the two particles be \( v_1 \) and \( v_2 \).
- The rate at which the distance decreases is given by:
\[ v_1 + v_2 = 6 \quad \text{(1)} \]

2. **When the particles are moving in the same direction:**
- The rate at which the distance increases is given by:
\[ v_1 - v_2 = 4 \quad \text{(2)} \]

### Step 3: Solve the equations
Now we have a system of two equations:
1. \( v_1 + v_2 = 6 \)
2. \( v_1 - v_2 = 4 \)

We can solve these equations simultaneously.

### Step 4: Add the equations
Adding equations (1) and (2):
\[ (v_1 + v_2) + (v_1 - v_2) = 6 + 4 \]
This simplifies to:
\[ 2v_1 = 10 \]
So,
\[ v_1 = 5 \, \text{m/s} \]

### Step 5: Substitute to find \( v_2 \)
Now substitute \( v_1 = 5 \) into equation (1):
\[ 5 + v_2 = 6 \]
This gives:
\[ v_2 = 1 \, \text{m/s} \]

### Conclusion
The speeds of the two particles are:
- \( v_1 = 5 \, \text{m/s} \)
- \( v_2 = 1 \, \text{m/s} \)

### Final Answer
The particles have speeds of 5 m/s and 1 m/s.

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