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 parts with different velocities, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let's assume the total distance of the journey is \( D \). According to the problem, the journey is divided into three equal parts, so each part will be \( \frac{D}{3} \).

2. **Identify the Velocities**:
- For the first \( \frac{D}{3} \) part, the velocity \( v_1 = 2 \, \text{m/s} \).
- For the second \( \frac{D}{3} \) part, the velocity \( v_2 = 3 \, \text{m/s} \).
- For the last \( \frac{D}{3} \) part, the velocity \( v_3 = 6 \, \text{m/s} \).

3. **Calculate the Time Taken for Each Part**:
- The time taken for the first part \( t_1 \):
\[ t_1 = \frac{\text{Distance}}{\text{Velocity}} = \frac{\frac{D}{3}}{2} = \frac{D}{6} \, \text{s} \]
- The time taken for the second part \( t_2 \):
\[ t_2 = \frac{\frac{D}{3}}{3} = \frac{D}{9} \, \text{s} \]
- The time taken for the third part \( t_3 \):
\[ t_3 = \frac{\frac{D}{3}}{6} = \frac{D}{18} \, \text{s} \]

4. **Calculate the Total Time Taken**:
The total time \( T \) taken for the entire journey is the sum of the times for each part:
\[ T = t_1 + t_2 + t_3 = \frac{D}{6} + \frac{D}{9} + \frac{D}{18} \]
To add these fractions, we need a common denominator. The least common multiple of 6, 9, and 18 is 18:
\[ T = \frac{3D}{18} + \frac{2D}{18} + \frac{D}{18} = \frac{6D}{18} = \frac{D}{3} \, \text{s} \]

5. **Calculate the Average Velocity**:
The average velocity \( V_{avg} \) is given by the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{D}{3}} = 3 \, \text{m/s} \]

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

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