A car completes the first half of its journey with a velocity `V_(1)` and the remaining half with a velocity `V_(2)`. Then the average velocity of the car for the whole journey is

Options

A. `(V_(1) + V_(2))/(2)`
B. `sqrt(V_(1) + V_(2))`
C. `(2V_(1)V_(2))/(V_(1) + V_(2))`
D. `(1)/(V_(1)) + (1)/(V_(2))`

Answer

To find the average velocity of a car that completes the first half of its journey with velocity \( V_1 \) and the second half with velocity \( V_2 \), we can use the concept of harmonic mean. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Understanding the Journey**:
- The car travels two equal distances, let's denote each half of the journey as \( d \). Therefore, the total distance \( D \) is \( 2d \).

2. **Calculating Time for Each Half**:
- The time taken to cover the first half of the journey (distance \( d \) at velocity \( V_1 \)) is given by:
\[ t_1 = \frac{d}{V_1} \]
- The time taken to cover the second half of the journey (distance \( d \) at velocity \( V_2 \)) is given by:
\[ t_2 = \frac{d}{V_2} \]

3. **Total Time for the Journey**:
- The total time \( T \) for the entire journey is the sum of the times for each half:
\[ T = t_1 + t_2 = \frac{d}{V_1} + \frac{d}{V_2} \]

4. **Finding the Average Velocity**:
- The average velocity \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T} = \frac{2d}{\frac{d}{V_1} + \frac{d}{V_2}} \]
- Simplifying this expression, we can factor out \( d \):
\[ V_{avg} = \frac{2d}{d\left(\frac{1}{V_1} + \frac{1}{V_2}\right)} = \frac{2}{\frac{1}{V_1} + \frac{1}{V_2}} \]

5. **Using the Harmonic Mean**:
- The expression \( \frac{2}{\frac{1}{V_1} + \frac{1}{V_2}} \) is the formula for the harmonic mean of \( V_1 \) and \( V_2 \):
\[ V_{avg} = \frac{2V_1 V_2}{V_1 + V_2} \]

6. **Conclusion**:
- Therefore, the average velocity of the car for the whole journey is:
\[ V_{avg} = \frac{2V_1 V_2}{V_1 + V_2} \]

### Final Answer:
The average velocity of the car for the whole journey is \( \frac{2V_1 V_2}{V_1 + V_2} \).