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:-

Explanation

To find the average velocity of a body that covers one-third of the distance with different velocities, we can follow these steps:

### Step 1: Define the Total Distance
Let the total distance covered by the body be \( D \). Since the body covers one-third of the distance three times, we can express the total distance as:
\[ D_{total} = D + D + D = 3D \]

### Step 2: Calculate the Time Taken for Each Segment
The body covers each one-third of the distance with different velocities. The time taken to cover each segment can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} \]

- For the first one-third of the distance \( D \) at velocity \( v_1 \):
\[ T_1 = \frac{D}{v_1} \]

- For the second one-third of the distance \( D \) at velocity \( v_2 \):
\[ T_2 = \frac{D}{v_2} \]

- For the last one-third of the distance \( D \) at velocity \( v_3 \):
\[ T_3 = \frac{D}{v_3} \]

### Step 3: Calculate the Total Time Taken
The total time taken to cover the entire distance is the sum of the times for each segment:
\[ T_{total} = T_1 + T_2 + T_3 = \frac{D}{v_1} + \frac{D}{v_2} + \frac{D}{v_3} \]

### Step 4: Factor Out the Common Distance
We can factor out \( D \) from the total time:
\[ T_{total} = D \left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right) \]

### Step 5: Calculate the Average Velocity
The average velocity \( V_{avg} \) is defined as the total displacement divided by the total time taken:
\[ V_{avg} = \frac{D_{total}}{T_{total}} = \frac{3D}{D \left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} \]

### Step 6: Simplify the Expression
Cancelling \( D \) from the numerator and denominator gives:
\[ V_{avg} = \frac{3}{\left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} \]

### Step 7: Final Result
To express the average velocity in a more standard form, we can rewrite it as:
\[ V_{avg} = \frac{3v_1 v_2 v_3}{v_2 v_3 + v_1 v_3 + v_1 v_2} \]

### Summary
Thus, the average velocity of the body is:
\[ V_{avg} = \frac{3v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_1 v_3} \]