A man walking briskly in rain with speed v must slant his umbrella forward making an angle `theta` with the vertical. A student derives the following relation between `theta` and v as tan `theta=v` and checks that the relation has a correct limit when `vto0,thetato0` as expected. Do you think the relation can be correct? If not guess out the correct relation.

Answer

To solve the problem, we need to analyze the relationship between the angle \(\theta\) that the umbrella makes with the vertical and the speed \(v\) of the man walking in the rain. The initial claim is that \(\tan \theta = v\), but we need to check if this is dimensionally correct and if it makes sense in the context of the problem.

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- A man is walking in the rain with speed \(v\).
- He tilts his umbrella at an angle \(\theta\) with respect to the vertical to avoid getting wet.

2. **Analyzing the Forces**:
- The rain is falling vertically with some speed \(v_r\).
- The man is moving horizontally with speed \(v\).
- The resultant velocity of the rain relative to the man will be a combination of these two velocities.

3. **Finding the Resultant Velocity**:
- The horizontal component of the rain's velocity is \(v\) (the man's speed).
- The vertical component of the rain's velocity is \(v_r\) (the speed of rain falling).
- The angle \(\theta\) can be defined using the tangent function:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{v}{v_r} \]

4. **Checking the Given Relation**:
- The student derived the relation \(\tan \theta = v\).
- This is dimensionally incorrect because \(\tan \theta\) is dimensionless (it has no units), while \(v\) has dimensions of \([L][T^{-1}]\).
- Therefore, the relation \(\tan \theta = v\) cannot be correct.

5. **Finding the Correct Relation**:
- From our analysis, we found that the correct relation should be:
\[ \tan \theta = \frac{v}{v_r} \]
- This indicates that the angle \(\theta\) depends on the ratio of the man's speed to the speed of the rain.

6. **Limit Check**:
- As \(v \to 0\) (the man is standing still), we expect \(\theta \to 0\) if \(v_r\) is non-zero.
- This limit holds true for the derived relation \(\tan \theta = \frac{v}{v_r}\).

### Conclusion:
The relation \(\tan \theta = v\) is incorrect. The correct relation is:
\[ \tan \theta = \frac{v}{v_r} \]