A person travelling on a straight line moves with a uniform velocity `v_1 for a distance x and with a uniform velocity `v_2` 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 \( v \) of a person who travels equal distances \( x \) with two different uniform velocities \( v_1 \) and \( v_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total distance traveled**:
The person travels a distance \( x \) at velocity \( v_1 \) and then another distance \( x \) at velocity \( v_2 \). Therefore, the total distance \( D \) is:
\[ D = x + x = 2x \]

2. **Calculate the time taken for each segment**:
- For the first segment (distance \( x \) at velocity \( v_1 \)):
\[ t_1 = \frac{x}{v_1} \]
- For the second segment (distance \( x \) at velocity \( v_2 \)):
\[ t_2 = \frac{x}{v_2} \]

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

4. **Factor out \( x \)**:
We can factor out \( x \) from the total time equation:
\[ T = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

5. **Calculate the average velocity**:
The average velocity \( v \) is defined as the total distance divided by the total time:
\[ v = \frac{D}{T} = \frac{2x}{T} \]
Substituting \( T \) from the previous step:
\[ v = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \]

6. **Simplify the expression**:
The average velocity can be expressed as:
\[ \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} = \frac{2v_1v_2}{v_1 + v_2} \]

### Final Result:
Thus, the average velocity \( v \) is given by:
\[ v = \frac{2v_1v_2}{v_1 + v_2} \]