Two men `Pa n dQ` start with velocity `u` at the same time from the junction of two roads inclined at `45^0` to each other. If they travel by different roads, find the rate at which they are being separated.

Answer

To solve the problem, we need to find the rate at which two men, P and Q, are separating as they travel along two roads inclined at an angle of \(45^\circ\) to each other, both starting from the same point with the same velocity \(u\).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let \(O\) be the junction where both men start.
- Let \(P\) travel along one road and \(Q\) travel along the other road.
- The angle between the two roads is \(45^\circ\).

2. **Setting Up the Variables**:
- Let \(x\) be the distance traveled by each man after time \(t\).
- Since both men travel with the same velocity \(u\), the distance \(x\) can be expressed as:
\[ x = ut \]
- Let \(y\) be the distance between the two men after time \(t\).

3. **Using the Law of Cosines**:
- According to the law of cosines, for the triangle formed by the points \(O\), \(P\), and \(Q\):
\[ y^2 = x^2 + x^2 - 2x^2 \cos(45^\circ) \]
- Since \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\), we can substitute this into the equation:
\[ y^2 = 2x^2 - 2x^2 \cdot \frac{1}{\sqrt{2}} \]
- Simplifying this gives:
\[ y^2 = 2x^2(1 - \frac{1}{\sqrt{2}}) \]

4. **Simplifying Further**:
- Let’s factor out \(x^2\):
\[ y^2 = 2x^2 \left(1 - \frac{1}{\sqrt{2}}\right) \]
- We can express \(y\) as:
\[ y = x \sqrt{2 \left(1 - \frac{1}{\sqrt{2}}\right)} \]

5. **Finding the Rate of Change**:
- To find \(\frac{dy}{dt}\), we differentiate both sides with respect to \(t\):
\[ \frac{dy}{dt} = \sqrt{2 \left(1 - \frac{1}{\sqrt{2}}\right)} \cdot \frac{dx}{dt} \]
- Since \(x = ut\), we have \(\frac{dx}{dt} = u\):
\[ \frac{dy}{dt} = \sqrt{2 \left(1 - \frac{1}{\sqrt{2}}\right)} \cdot u \]

6. **Final Calculation**:
- We can calculate \(1 - \frac{1}{\sqrt{2}} = \frac{\sqrt{2} - 1}{\sqrt{2}}\), so:
\[ \frac{dy}{dt} = u \sqrt{2 \cdot \frac{\sqrt{2} - 1}{\sqrt{2}}} = u \sqrt{2(\sqrt{2} - 1)} \]

### Final Result:
The rate at which the two men are separating is given by:
\[ \frac{dy}{dt} = u \sqrt{2(\sqrt{2} - 1)} \]