A vehicle travels half the distance (L) with speed ` V_1` and the other half with speed ` V_2`, then its average speed is .

Options

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

Explanation

To find the average speed of a vehicle that travels half the distance \( L \) with speed \( V_1 \) and the other half with speed \( V_2 \), we can follow these steps:

### Step 1: Define the total distance
The total distance \( L \) is divided into two equal halves. Therefore, each half is:
\[ \text{Distance for each half} = \frac{L}{2} \]

### Step 2: Calculate the time taken for each half
1. **Time for the first half** (with speed \( V_1 \)):
\[ T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{L}{2}}{V_1} = \frac{L}{2V_1} \]

2. **Time for the second half** (with speed \( V_2 \)):
\[ T_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{L}{2}}{V_2} = \frac{L}{2V_2} \]

### Step 3: Calculate the total time taken
The total time \( T \) taken for the entire journey is the sum of the times for both halves:
\[ T = T_1 + T_2 = \frac{L}{2V_1} + \frac{L}{2V_2} \]

### Step 4: Simplify the total time
We can factor out \( \frac{L}{2} \) from the total time:
\[ T = \frac{L}{2} \left( \frac{1}{V_1} + \frac{1}{V_2} \right) \]

### Step 5: Calculate the average speed
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{L}{T} \]
Substituting the expression for \( T \):
\[ V_{avg} = \frac{L}{\frac{L}{2} \left( \frac{1}{V_1} + \frac{1}{V_2} \right)} = \frac{L \cdot 2}{L \left( \frac{1}{V_1} + \frac{1}{V_2} \right)} \]
This simplifies to:
\[ V_{avg} = \frac{2}{\frac{1}{V_1} + \frac{1}{V_2}} \]

### Step 6: Final expression for average speed
Using the formula for the sum of reciprocals, we can write:
\[ V_{avg} = \frac{2 V_1 V_2}{V_1 + V_2} \]

Thus, the average speed of the vehicle is:
\[ \boxed{\frac{2 V_1 V_2}{V_1 + V_2}} \]