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. ` (2 V_1 + V_2)/(V_1 +V_2)`
C. ` ( 2 V_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: Understand the total distance
The total distance \( L \) is divided into two equal halves. Therefore, the distance for each half is:
\[ \text{Distance for each half} = \frac{L}{2} \]

### Step 2: Calculate the time taken for each half
The time taken to cover the first half of the distance with speed \( V_1 \) can be calculated using the formula:
\[ T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{L}{2}}{V_1} = \frac{L}{2V_1} \]

Similarly, the time taken to cover the second half of the distance with speed \( V_2 \) is:
\[ T_2 = \frac{\frac{L}{2}}{V_2} = \frac{L}{2V_2} \]

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

### Step 4: Simplify the total time expression
To combine the two fractions, we can find a common denominator:
\[ T = \frac{L}{2} \left( \frac{1}{V_1} + \frac{1}{V_2} \right) \]
This can be rewritten as:
\[ T = \frac{L}{2} \cdot \frac{V_1 + V_2}{V_1 V_2} = \frac{L(V_1 + V_2)}{2V_1 V_2} \]

### 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 \( T \):
\[ V_{avg} = \frac{L}{\frac{L(V_1 + V_2)}{2V_1 V_2}} = \frac{L \cdot 2V_1 V_2}{L(V_1 + V_2)} \]
The \( L \) cancels out:
\[ V_{avg} = \frac{2V_1 V_2}{V_1 + V_2} \]

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