Rain is falling at the speed of ` 25sqrt3 m//s` vertically. The wind blows west to east at a speed of 25 m/s. find the velocity of rain as experienced by a person standing on the ground.

Answer

To find the velocity of rain as experienced by a person standing on the ground, we can break down the problem into steps involving vector addition.

### Step 1: Identify the velocities
- The velocity of the rain falling vertically downward is given as \( v_r = -25\sqrt{3} \, \text{m/s} \) (negative because it is downward).
- The velocity of the wind blowing from west to east is given as \( v_w = 25 \, \text{m/s} \) (positive because it is in the eastward direction).

### Step 2: Represent the velocities as vectors
- The velocity of rain can be represented as a vector:
\[ \vec{v_r} = 0 \hat{i} - 25\sqrt{3} \hat{j} \]
- The velocity of wind can be represented as a vector:
\[ \vec{v_w} = 25 \hat{i} + 0 \hat{j} \]

### Step 3: Calculate the resultant velocity
To find the resultant velocity of the rain as experienced by a person on the ground, we add the two vectors:
\[ \vec{v} = \vec{v_w} + \vec{v_r} = (25 \hat{i} + 0 \hat{j}) + (0 \hat{i} - 25\sqrt{3} \hat{j}) = 25 \hat{i} - 25\sqrt{3} \hat{j} \]

### Step 4: Calculate the magnitude of the resultant velocity
The magnitude of the resultant velocity vector can be calculated using the Pythagorean theorem:
\[ |\vec{v}| = \sqrt{(25)^2 + (-25\sqrt{3})^2} \]
Calculating each term:
- \( (25)^2 = 625 \)
- \( (-25\sqrt{3})^2 = 625 \cdot 3 = 1875 \)

Now, adding these:
\[ |\vec{v}| = \sqrt{625 + 1875} = \sqrt{2500} = 50 \, \text{m/s} \]

### Final Answer
The velocity of the rain as experienced by a person standing on the ground is \( 50 \, \text{m/s} \).

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