A particle travels first half of the total time with speed `v_1` and second half time with speed `v_2.` Find the average speed during the complete journey.

Explanation

To find the average speed of a particle that travels the first half of the total time with speed \( v_1 \) and the second half with speed \( v_2 \), we can follow these steps:

### Step 1: Define Total Time
Let the total time of the journey be \( T \). For simplicity, we can express \( T \) as \( 2t \), where \( t \) is half of the total time. Thus, the first half of the journey lasts for \( t \) and the second half also lasts for \( t \).

### Step 2: Calculate Distance for Each Half
1. **Distance during the first half**: The distance traveled during the first half of the time with speed \( v_1 \) is given by:
\[ d_1 = v_1 \cdot t \]
2. **Distance during the second half**: The distance traveled during the second half of the time with speed \( v_2 \) is:
\[ d_2 = v_2 \cdot t \]

### Step 3: Calculate Total Distance
The total distance \( D \) traveled during the entire journey is the sum of the distances from both halves:
\[ D = d_1 + d_2 = v_1 \cdot t + v_2 \cdot t = t(v_1 + v_2) \]

### Step 4: Calculate Total Time
The total time for the journey is:
\[ T = 2t \]

### Step 5: Calculate Average Speed
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} = \frac{t(v_1 + v_2)}{2t} \]
Simplifying this gives:
\[ V_{avg} = \frac{v_1 + v_2}{2} \]

### Final Answer
Thus, the average speed during the complete journey is:
\[ V_{avg} = \frac{v_1 + v_2}{2} \]

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