A man walks for some time 't' with velocity(v) due east. Then he walks for same time 't' with velocity (v) due north. The average velocity of the man is:-

Options

A. `(2v_(0)^(2))/(g)`
B. `v_(0)^(2)/g`
C. `v_(0)^(2)/(2g)`
D. `v_(0)^(2)/(4g)`

Explanation

To find the average velocity of a man who walks due east and then due north, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Directions and Velocities**:
- The man walks due east with velocity \( v \) for time \( t \).
- Then, he walks due north with the same velocity \( v \) for the same time \( t \).

2. **Calculate the Displacement for Each Segment**:
- For the first segment (due east):
\[ \text{Displacement}_1 = v \cdot t \quad \text{(in the east direction)} \]
- For the second segment (due north):
\[ \text{Displacement}_2 = v \cdot t \quad \text{(in the north direction)} \]

3. **Express the Displacements in Vector Form**:
- The displacement vector for the eastward movement can be represented as:
\[ \vec{d_1} = vt \hat{i} \]
- The displacement vector for the northward movement can be represented as:
\[ \vec{d_2} = vt \hat{j} \]

4. **Calculate the Total Displacement**:
- The total displacement vector \( \vec{D} \) is the vector sum of the two displacements:
\[ \vec{D} = \vec{d_1} + \vec{d_2} = vt \hat{i} + vt \hat{j} \]

5. **Simplify the Total Displacement**:
- Factor out \( vt \):
\[ \vec{D} = vt (\hat{i} + \hat{j}) \]

6. **Calculate the Magnitude of the Total Displacement**:
- The magnitude of the total displacement \( D \) can be calculated using the Pythagorean theorem:
\[ D = \sqrt{(vt)^2 + (vt)^2} = \sqrt{2(vt)^2} = vt\sqrt{2} \]

7. **Calculate the Total Time**:
- The total time \( T \) taken for both segments is:
\[ T = t + t = 2t \]

8. **Calculate the Average Velocity**:
- The average velocity \( \vec{V}_{\text{avg}} \) is given by the formula:
\[ \vec{V}_{\text{avg}} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{vt\sqrt{2}}{2t} \]
- Simplifying this gives:
\[ \vec{V}_{\text{avg}} = \frac{v\sqrt{2}}{2} \]

9. **Final Result**:
- The average velocity of the man is:
\[ \vec{V}_{\text{avg}} = \frac{v}{\sqrt{2}} \quad \text{(in the direction of } \hat{i} + \hat{j}\text{)} \]