A person travelling on a straight line moves with a uniform velocity v1 for a distance x and with a uniform velocity v2 for the next equal distance. The average velocity v is given by

Options

A. `v=(v_1+v_2)/2`
B. `v=sqrt(v_1v_2)`
C. `2/v=1/v_1+1/v_2`
D. `1/v=1/v_1+1/v_2`

Answer

To find the average velocity of a person traveling with two different uniform velocities over equal distances, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Distances and Velocities**:
- The person travels a distance \( x \) with a velocity \( v_1 \).
- Then, the person travels another distance \( x \) with a velocity \( v_2 \).

2. **Calculate the Time Taken for Each Segment**:
- The time taken to travel the first distance \( x \) at velocity \( v_1 \) is given by:
\[ t_1 = \frac{x}{v_1} \]
- The time taken to travel the second distance \( x \) at velocity \( v_2 \) is given by:
\[ t_2 = \frac{x}{v_2} \]

3. **Calculate the Total Displacement**:
- The total displacement for the entire journey is:
\[ \text{Total Displacement} = x + x = 2x \]

4. **Calculate the Total Time Taken**:
- The total time taken for the journey is:
\[ \text{Total Time} = t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2} \]

5. **Combine the Times**:
- Factor out \( x \) from the total time:
\[ \text{Total Time} = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

6. **Calculate the Average Velocity**:
- The average velocity \( v \) is defined as total displacement divided by total time:
\[ v = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]
- Simplifying this expression gives:
\[ v = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \]

7. **Final Expression for Average Velocity**:
- The expression can be rewritten using the formula for the harmonic mean:
\[ v = \frac{2 v_1 v_2}{v_1 + v_2} \]

### Final Answer:
The average velocity \( v \) is given by:
\[ v = \frac{2 v_1 v_2}{v_1 + v_2} \]