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A person walking at 4 m/s finds rain dro...

A person walking at 4 m/s finds rain drops falling slantwise into his face with a speed of 4 m/s at an angle of `30^(@)` with the vertical . Show that the actual speed of the rain drops is 4 m/s .

A

4 m/s

B

8 m/s

C

6 m/s

D

5 m/s

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the actual speed of the raindrops falling at an angle of 30 degrees with respect to the vertical while a person walks at a speed of 4 m/s. ### Step-by-Step Solution: 1. **Understand the Given Information:** - The speed of the person (V_pg) = 4 m/s (horizontal). - The speed of the raindrops relative to the person (V_rp) = 4 m/s (at an angle of 30 degrees with the vertical). 2. **Determine the Components of the Rain's Velocity Relative to the Person:** - Since the raindrops are falling at an angle of 30 degrees with the vertical, we can break this velocity into its vertical and horizontal components. - The vertical component (V_rp_vertical) = V_rp * cos(30°) = 4 * (√3/2) = 2√3 m/s. - The horizontal component (V_rp_horizontal) = V_rp * sin(30°) = 4 * (1/2) = 2 m/s. 3. **Set Up the Velocity of the Rain with Respect to the Ground:** - The actual velocity of the rain with respect to the ground (V_rg) can be found using the vector addition of the rain's velocity relative to the person and the person's velocity. - V_rg_vertical = V_rp_vertical (2√3 m/s). - V_rg_horizontal = V_rp_horizontal - V_pg = 2 m/s - 4 m/s = -2 m/s (indicating it is falling towards the opposite direction). 4. **Calculate the Magnitude of the Actual Velocity of the Rain:** - The magnitude of the actual velocity of the rain can be calculated using the Pythagorean theorem: \[ |V_rg| = \sqrt{(V_rg_horizontal)^2 + (V_rg_vertical)^2} \] \[ |V_rg| = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4 \text{ m/s} \] 5. **Conclusion:** - The actual speed of the raindrops is 4 m/s.
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