A body covers first `1/3` part of its journey with a velocity of `2m//s`, next `1/3` part with a velocity of `(3m)/s` and rest of the journey with a velocity `6m//s`. The average velocity of the body will be

Options

A. 3m/s
B. 11/3 m/s
C. 8/3 m/s
D. 4/3 m/s

Explanation

To find the average velocity of the body that covers its journey in three segments with different velocities, we can follow these steps:

### Step 1: Define the total distance
Let the total distance of the journey be \( x \). Since the journey is divided into three equal parts, each part will be \( \frac{x}{3} \).

### Step 2: Calculate time taken for each segment
1. **First Segment**:
- Distance = \( \frac{x}{3} \)
- Velocity = \( 2 \, \text{m/s} \)
- Time taken \( t_1 = \frac{\text{Distance}}{\text{Velocity}} = \frac{\frac{x}{3}}{2} = \frac{x}{6} \)

2. **Second Segment**:
- Distance = \( \frac{x}{3} \)
- Velocity = \( 3 \, \text{m/s} \)
- Time taken \( t_2 = \frac{\frac{x}{3}}{3} = \frac{x}{9} \)

3. **Third Segment**:
- Distance = \( \frac{x}{3} \)
- Velocity = \( 6 \, \text{m/s} \)
- Time taken \( t_3 = \frac{\frac{x}{3}}{6} = \frac{x}{18} \)

### Step 3: Calculate total time taken
Now, we can find the total time taken for the entire journey by summing up the time taken for each segment:
\[ \text{Total time} = t_1 + t_2 + t_3 = \frac{x}{6} + \frac{x}{9} + \frac{x}{18} \]

To add these fractions, we need a common denominator. The least common multiple of \( 6, 9, \) and \( 18 \) is \( 18 \):
\[ \frac{x}{6} = \frac{3x}{18}, \quad \frac{x}{9} = \frac{2x}{18}, \quad \frac{x}{18} = \frac{x}{18} \]
So,
\[ \text{Total time} = \frac{3x}{18} + \frac{2x}{18} + \frac{x}{18} = \frac{6x}{18} = \frac{x}{3} \]

### Step 4: Calculate average velocity
The average velocity \( v_{\text{avg}} \) is given by the formula:
\[ v_{\text{avg}} = \frac{\text{Total displacement}}{\text{Total time}} \]
Since the total displacement is \( x \) and the total time is \( \frac{x}{3} \):
\[ v_{\text{avg}} = \frac{x}{\frac{x}{3}} = 3 \, \text{m/s} \]

### Final Answer
The average velocity of the body is \( 3 \, \text{m/s} \).

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