A particle travels first half of the total distance with constant speed `v_1` and second half with constant 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 a total distance with constant speed \( v_1 \) and the second half with constant speed \( v_2 \), we can follow these steps:

### Step 1: Define the total distance
Let the total distance traveled by the particle be \( D \). For simplification, we can assume \( D = 2x \), where \( x \) is the distance for each half of the journey. Thus, the first half of the distance is \( x \) and the second half is also \( x \).

### Step 2: Calculate the time taken for each half
- **Time for the first half (t1)**: The time taken to travel the first half of the distance \( x \) at speed \( v_1 \) is given by:
\[ t_1 = \frac{x}{v_1} \]

- **Time for the second half (t2)**: The time taken to travel the second half of the distance \( x \) at speed \( v_2 \) is given by:
\[ t_2 = \frac{x}{v_2} \]

### Step 3: Calculate the total time taken
The total time \( T \) taken for the entire journey is the sum of the times for each half:
\[ T = t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2} \]

### Step 4: Simplify the total time expression
We can factor out \( x \) from the total time expression:
\[ T = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

### Step 5: Find the average speed
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} \]
Substituting \( D = 2x \) and \( T = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \):
\[ V_{avg} = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]

### Step 6: Cancel \( x \) and simplify
The \( x \) terms cancel out:
\[ V_{avg} = \frac{2}{\left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]

### Step 7: Final expression for average speed
To express this in a more standard form, we can rewrite it as:
\[ V_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} \]

Thus, the average speed during the complete journey is:
\[ \boxed{\frac{2 v_1 v_2}{v_1 + v_2}} \]

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