A man is going due east with a velocity of `3 kmh^(-1)`. Rain falls vertically downwards at a speed of `10 km h^(-1)`. Calculate the angle at which he should hold his umbrella so as to protect himself from rain.

Answer

To solve the problem of determining the angle at which a man should hold his umbrella to protect himself from the rain, we can follow these steps:

### Step 1: Understand the velocities involved
- The man is moving due east with a velocity of \(3 \, \text{km/h}\).
- The rain is falling vertically downwards with a velocity of \(10 \, \text{km/h}\).

### Step 2: Set up the coordinate system
- Let the east direction be along the positive x-axis.
- Let the vertical downward direction be along the negative y-axis.

### Step 3: Define the velocity vectors
- The velocity of the man can be represented as:
\[ \vec{V}_{\text{man}} = 3 \hat{i} \, \text{km/h} \]
- The velocity of the rain can be represented as:
\[ \vec{V}_{\text{rain}} = -10 \hat{j} \, \text{km/h} \]

### Step 4: Calculate the relative velocity of rain with respect to the man
- The velocity of rain with respect to the man is given by:
\[ \vec{V}_{\text{rain, man}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{man}} \]
Substituting the values:
\[ \vec{V}_{\text{rain, man}} = (-10 \hat{j}) - (3 \hat{i}) = -3 \hat{i} - 10 \hat{j} \]

### Step 5: Determine the angle of the relative velocity vector
- The angle \(\alpha\) at which the umbrella should be held can be found using the tangent function:
\[ \tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{|\text{velocity in x}|}{|\text{velocity in y}|} = \frac{3}{10} \]

### Step 6: Calculate the angle \(\alpha\)
- To find \(\alpha\), we take the arctangent:
\[ \alpha = \tan^{-1}\left(\frac{3}{10}\right) \]

### Step 7: Interpret the angle
- The angle \(\alpha\) is measured from the vertical towards the east. Thus, the man should hold his umbrella at an angle of \(\tan^{-1}(0.3)\) east of vertical.

### Final Answer
The man should hold his umbrella at an angle of \(\tan^{-1}(0.3)\) east of vertical.

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