A particle travels first half of the total distance with speed `v_1.` In second half distace with speed in1/3 rd timeis `v_2.` and in remaining 2/3 rd time constant speed is `v_3.` Find the average speed during the complete journey.

Explanation

To find the average speed during the complete journey of the particle, we will follow these steps:

### Step 1: Define the total distance
Let the total distance traveled by the particle be \( 2d \). This means the first half of the journey is \( d \) and the second half is also \( d \).

### Step 2: Calculate the time taken for the first half of the journey
The particle travels the first half of the distance \( d \) with speed \( v_1 \). The time taken for this part can be calculated using the formula:
\[ t_1 = \frac{d}{v_1} \]

### Step 3: Calculate the time taken for the second half of the journey
In the second half of the journey, the particle travels the distance \( d \) but with speed \( v_2 \) in \( \frac{1}{3} \) of the time taken for the first half. Therefore, the time taken for this part is:
\[ t_2 = \frac{1}{3} t_1 = \frac{1}{3} \left( \frac{d}{v_1} \right) = \frac{d}{3v_1} \]

### Step 4: Calculate the time taken for the remaining distance
After the second half, the particle travels for the remaining \( \frac{2}{3} \) of the time at a constant speed \( v_3 \). The total time for the journey so far is:
\[ t_{\text{total}} = t_1 + t_2 = \frac{d}{v_1} + \frac{d}{3v_1} = \frac{3d + d}{3v_1} = \frac{4d}{3v_1} \]
The remaining time is:
\[ t_3 = \frac{2}{3} \left( \frac{4d}{3v_1} \right) = \frac{8d}{9v_1} \]

### Step 5: Calculate the distance covered in the remaining time
The distance covered in this remaining time \( t_3 \) at speed \( v_3 \) is:
\[ d_3 = v_3 \cdot t_3 = v_3 \cdot \frac{8d}{9v_1} \]

### Step 6: Set up the equation for total distance
Since the total distance is \( 2d \), we can set up the equation:
\[ d + d_3 = 2d \]
Substituting \( d_3 \):
\[ d + \frac{8d v_3}{9v_1} = 2d \]
This simplifies to:
\[ \frac{8d v_3}{9v_1} = d \]
Thus,
\[ 8v_3 = 9v_1 \implies v_3 = \frac{9}{8}v_1 \]

### Step 7: Calculate total time
Now we can find the total time taken for the journey:
\[ T = t_1 + t_2 + t_3 = \frac{d}{v_1} + \frac{d}{3v_1} + \frac{8d}{9v_1} \]
Finding a common denominator (which is 9):
\[ T = \frac{9d}{9v_1} + \frac{3d}{9v_1} + \frac{8d}{9v_1} = \frac{20d}{9v_1} \]

### Step 8: Calculate average speed
The average speed \( V_{\text{avg}} \) is given by:
\[ V_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{T} = \frac{2d}{\frac{20d}{9v_1}} = \frac{2d \cdot 9v_1}{20d} = \frac{18v_1}{20} = \frac{9v_1}{10} \]

### Final Answer
Thus, the average speed during the complete journey is:
\[ \boxed{\frac{9v_1}{10}} \]