A body covers one-third of the distance with a velocity `v_(1)` the second one-third of the distance with a velocity `v_(2)`, and the last one-third of the distance with a velocity `v_(3)`. The average velocity is:-

Options

A. `(v_(1)+v_(2)+v_(3))/(3)`
B. `(3v_(1)v_(2)v_(3))/(v_(1)v_(2)+v_(2)v_(3)+v_(3)v_(1))`
C. `(v_(1)v_(2)+v_(2)v_(3)+v_(3)v_(1))/(3)`
D. `(v_(1)v_(2)v_(3))/(3)`

Answer

To find the average velocity of a body that covers one-third of the distance with a velocity \( v_1 \), the second one-third with a velocity \( v_2 \), and the last one-third with a velocity \( v_3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let the total distance be \( D \). Since the body covers one-third of the distance in each segment, we can express the distance for each segment as:
\[ d_1 = d_2 = d_3 = \frac{D}{3} \]

2. **Calculate the Time for Each Segment**:
The time taken to cover each segment can be calculated using the formula:
\[ t = \frac{\text{distance}}{\text{velocity}} \]
Thus, for each segment:
- Time for the first segment \( t_1 \):
\[ t_1 = \frac{d_1}{v_1} = \frac{\frac{D}{3}}{v_1} = \frac{D}{3v_1} \]
- Time for the second segment \( t_2 \):
\[ t_2 = \frac{d_2}{v_2} = \frac{\frac{D}{3}}{v_2} = \frac{D}{3v_2} \]
- Time for the third segment \( t_3 \):
\[ t_3 = \frac{d_3}{v_3} = \frac{\frac{D}{3}}{v_3} = \frac{D}{3v_3} \]

3. **Calculate the Total Time**:
The total time \( T \) taken to cover the entire distance \( D \) is the sum of the times for each segment:
\[ T = t_1 + t_2 + t_3 = \frac{D}{3v_1} + \frac{D}{3v_2} + \frac{D}{3v_3} \]
Factor out \( \frac{D}{3} \):
\[ T = \frac{D}{3} \left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right) \]

4. **Calculate the Average Velocity**:
The average velocity \( V_{avg} \) is defined as the total displacement divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T} \]
Substituting \( T \):
\[ V_{avg} = \frac{D}{\frac{D}{3} \left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} = \frac{3}{\left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} \]

5. **Final Result**:
The average velocity can also be expressed using the formula for the harmonic mean of the velocities:
\[ V_{avg} = \frac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1} \]

### Conclusion:
The average velocity of the body is given by:
\[ V_{avg} = \frac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1} \]