Rain drops are felling with velocity `(2hat i- 3 hatj)` m/s.
What should be the velocity of man so rain drops hit him with speed 5 m/s.

Options

A. `6 hat i`
B. `-hati `
C. `- 2 hati `
D. Both (A) and (C)

Answer

To solve the problem, we need to find the velocity of the man such that the rain drops hit him with a speed of 5 m/s. The velocity of the rain is given as \( \vec{v}_{\text{rain}} = 2\hat{i} - 3\hat{j} \) m/s.

### Step-by-Step Solution:

1. **Understand the Velocity of Rain**:
The rain is falling with a velocity vector \( \vec{v}_{\text{rain}} = 2\hat{i} - 3\hat{j} \). Here, \( 2\hat{i} \) represents the horizontal component and \( -3\hat{j} \) represents the vertical component.

2. **Define the Velocity of the Man**:
Let the velocity of the man be \( \vec{v}_{\text{man}} = v_x \hat{i} + v_y \hat{j} \).

3. **Relative Velocity of Rain with Respect to Man**:
The relative velocity of rain with respect to the man is given by:
\[ \vec{v}_{\text{relative}} = \vec{v}_{\text{rain}} - \vec{v}_{\text{man}} = (2 - v_x)\hat{i} + (-3 - v_y)\hat{j} \]

4. **Magnitude of Relative Velocity**:
We want the magnitude of this relative velocity to be 5 m/s:
\[ \sqrt{(2 - v_x)^2 + (-3 - v_y)^2} = 5 \]

5. **Square Both Sides**:
Squaring both sides gives:
\[ (2 - v_x)^2 + (-3 - v_y)^2 = 25 \]

6. **Expand the Equation**:
Expanding the equation:
\[ (2 - v_x)^2 = 4 - 4v_x + v_x^2 \]
\[ (-3 - v_y)^2 = 9 + 6v_y + v_y^2 \]
Combining these:
\[ 4 - 4v_x + v_x^2 + 9 + 6v_y + v_y^2 = 25 \]
Simplifying gives:
\[ v_x^2 + v_y^2 - 4v_x + 6v_y - 12 = 0 \]

7. **Rearranging the Equation**:
Rearranging the equation:
\[ v_x^2 + v_y^2 - 4v_x + 6v_y = 12 \]

8. **Solve for Specific Values**:
To find specific values for \( v_x \) and \( v_y \), we can assume \( v_y = -3 \) (to cancel out the vertical component of the rain). Substituting \( v_y = -3 \):
\[ v_x^2 - 4v_x + 6(-3) = 12 \]
\[ v_x^2 - 4v_x - 18 = 12 \]
\[ v_x^2 - 4v_x - 30 = 0 \]

9. **Using the Quadratic Formula**:
Using the quadratic formula \( v_x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ v_x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-30)}}{2 \cdot 1} \]
\[ v_x = \frac{4 \pm \sqrt{16 + 120}}{2} \]
\[ v_x = \frac{4 \pm \sqrt{136}}{2} \]
\[ v_x = \frac{4 \pm 2\sqrt{34}}{2} \]
\[ v_x = 2 \pm \sqrt{34} \]

10. **Final Values**:
Thus, the possible velocities of the man are:
\[ v_x = 2 + \sqrt{34} \quad \text{or} \quad v_x = 2 - \sqrt{34} \]