A person travelling on a straight line moves with a uniform velocity `v_1` for some time and with uniform velocity `v_2` for the next equal time. 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 with uniform velocities \( v_1 \) and \( v_2 \) for equal time intervals, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The person travels with velocity \( v_1 \) for time \( t \).
- Then, the person travels with velocity \( v_2 \) for the same time \( t \).

2. **Calculating Distances**:
- The distance covered during the first part of the journey (with velocity \( v_1 \)) is:
\[ x_1 = v_1 \cdot t \]
- The distance covered during the second part of the journey (with velocity \( v_2 \)) is:
\[ x_2 = v_2 \cdot t \]

3. **Total Displacement**:
- The total displacement \( x \) is the sum of the distances covered in both parts:
\[ x = x_1 + x_2 = v_1 \cdot t + v_2 \cdot t = (v_1 + v_2) \cdot t \]

4. **Total Time**:
- The total time \( T \) taken for the entire journey is:
\[ T = t + t = 2t \]

5. **Calculating Average Velocity**:
- The average velocity \( v \) is defined as the total displacement divided by the total time:
\[ v = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{x}{T} = \frac{(v_1 + v_2) \cdot t}{2t} \]
- Simplifying this expression:
\[ v = \frac{v_1 + v_2}{2} \]

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