Rain is falling vertically downwards with a velocity of 2m//s. A person is runing to the North with a velocity of `2sqrt(3)`m/s. Find out the velocity and direction of rain as appeared to the person.

Answer

To solve the problem of finding the velocity and direction of rain as perceived by a person running to the north, we can follow these steps:

### Step 1: Define the velocities
1. The velocity of the rain, \( \vec{V}_{\text{rain}} \), is given as \( 2 \, \text{m/s} \) vertically downwards. In vector form, this can be represented as:
\[ \vec{V}_{\text{rain}} = 0 \hat{i} + 0 \hat{j} - 2 \hat{k} \, \text{m/s} \]
(where \( \hat{k} \) is the unit vector in the vertical direction).

2. The velocity of the person running to the north is given as \( 2\sqrt{3} \, \text{m/s} \). In vector form, this can be represented as:
\[ \vec{V}_{\text{person}} = 0 \hat{i} + 2\sqrt{3} \hat{j} + 0 \hat{k} \, \text{m/s} \]
(where \( \hat{j} \) is the unit vector in the north direction).

### Step 2: Calculate the relative velocity of the rain with respect to the person
To find the velocity of the rain as perceived by the person, we need to subtract the person's velocity from the rain's velocity:
\[ \vec{V}_{\text{rain, relative}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{person}} \]
Substituting the values:
\[ \vec{V}_{\text{rain, relative}} = (0 \hat{i} + 0 \hat{j} - 2 \hat{k}) - (0 \hat{i} + 2\sqrt{3} \hat{j} + 0 \hat{k}) \]
This simplifies to:
\[ \vec{V}_{\text{rain, relative}} = 0 \hat{i} + (-2 - 2\sqrt{3}) \hat{j} + (-2) \hat{k} \]
Thus, we have:
\[ \vec{V}_{\text{rain, relative}} = 0 \hat{i} + (-2\sqrt{3}) \hat{j} - 2 \hat{k} \]

### Step 3: Calculate the magnitude of the relative velocity
The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V}_{\text{rain, relative}}| = \sqrt{(0)^2 + (-2\sqrt{3})^2 + (-2)^2} \]
Calculating the squares:
\[ |\vec{V}_{\text{rain, relative}}| = \sqrt{0 + 12 + 4} = \sqrt{16} = 4 \, \text{m/s} \]

### Step 4: Determine the direction of the relative velocity
To find the direction, we can express the relative velocity vector in terms of its components:
\[ \vec{V}_{\text{rain, relative}} = 0 \hat{i} + (-2\sqrt{3}) \hat{j} - 2 \hat{k} \]
The direction can be described in terms of angles or as a vector in the \( j-k \) plane. The angle \( \theta \) with respect to the vertical can be calculated using:
\[ \tan(\theta) = \frac{|\text{vertical component}|}{|\text{horizontal component}|} \]
Since the horizontal component is zero, the angle is simply downward.

### Final Answer
The velocity of the rain as perceived by the person is \( 4 \, \text{m/s} \) directed at an angle downward towards the vertical.

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