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 solve the problem of finding the average velocity of a person traveling with two different uniform velocities for equal time intervals, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
The person travels with a uniform velocity \( v_1 \) for a time \( t \) and then with a uniform velocity \( v_2 \) for the next equal time \( t \).

2. **Calculating Displacement for Each Segment**:
- For the first segment of the journey (with velocity \( v_1 \)):
\[ \text{Displacement } (s_1) = v_1 \times t \]
- For the second segment of the journey (with velocity \( v_2 \)):
\[ \text{Displacement } (s_2) = v_2 \times t \]

3. **Total Displacement**:
The total displacement \( S \) for the entire journey is the sum of the displacements from both segments:
\[ S = s_1 + s_2 = v_1 t + v_2 t = (v_1 + v_2) t \]

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

5. **Calculating Average Velocity**:
The average velocity \( v \) is defined as the total displacement divided by the total time:
\[ v = \frac{S}{T} = \frac{(v_1 + v_2) t}{2t} \]
Here, \( t \) cancels out:
\[ v = \frac{v_1 + v_2}{2} \]

6. **Final Result**:
Thus, the average velocity \( v \) is:
\[ v = \frac{v_1 + v_2}{2} \]