A person is walking to the east at 2 km/hr and the rain drops appear to him dropping vertically downwards at `2sqrt(3)km//hr`. Find the actual velocity of rain.

Answer

To find the actual velocity of the rain, we can analyze the situation using vector components. Here’s a step-by-step solution:

### Step 1: Define the velocities
- The person is walking to the east with a velocity of \( V_p = 2 \, \text{km/hr} \) in the positive x-direction.
- The rain appears to fall vertically downward at a velocity of \( V_{r/p} = 2\sqrt{3} \, \text{km/hr} \) with respect to the person.

### Step 2: Set up the equations
- The actual velocity of the rain \( V_r \) can be expressed as:
\[ V_r = V_{r/p} + V_p \]
- Here, \( V_{r/p} \) is the velocity of the rain with respect to the person, which is directed vertically downward (negative y-direction).

### Step 3: Express the velocities in vector form
- The velocity of the person can be represented as:
\[ V_p = 2 \hat{i} \, \text{km/hr} \]
- The velocity of the rain with respect to the person is:
\[ V_{r/p} = -2\sqrt{3} \hat{j} \, \text{km/hr} \]
- Therefore, the actual velocity of the rain can be expressed as:
\[ V_r = V_{r/p} + V_p = -2\sqrt{3} \hat{j} + 2 \hat{i} \]

### Step 4: Calculate the components of the actual velocity of the rain
- The actual velocity of the rain in vector form is:
\[ V_r = 2 \hat{i} - 2\sqrt{3} \hat{j} \]

### Step 5: Find the magnitude of the actual velocity
- The magnitude of the actual velocity \( |V_r| \) can be calculated using the Pythagorean theorem:
\[ |V_r| = \sqrt{(2)^2 + (-2\sqrt{3})^2} \]
\[ |V_r| = \sqrt{4 + 12} = \sqrt{16} = 4 \, \text{km/hr} \]

### Step 6: Determine the direction of the actual velocity
- To find the angle \( \theta \) that the velocity makes with the vertical, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{horizontal component}}{\text{vertical component}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} \]
- Therefore, \( \theta = 30^\circ \) with respect to the vertical.

### Final Answer
The actual velocity of the rain is \( 4 \, \text{km/hr} \) at an angle of \( 30^\circ \) with the vertical.

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