A man holds an umbrella at 30∘ with the vertical to keep himself dry. He, then, runs at a speed of 10 ms−1 and finds the rain drops to be hitting vertically. Speed of the rain drops w.r.t the running man and w.r.t earth are
Question
A man holds an umbrella at 30∘ with the vertical to keep himself dry. He, then, runs at a speed of 10 ms−1 and finds the rain drops to be hitting vertically. Speed of the rain drops w.r.t the running man and w.r.t earth are
Solution
The problem can be solved by using the concept of relative velocity.
Step 1: Understand the problem The man is holding the umbrella at an angle of 30 degrees to the vertical. This means that the rain is falling at an angle of 30 degrees to the vertical. When the man starts running, he perceives the rain to be falling vertically. We need to find the speed of the raindrops with respect to the man and with respect to the earth.
Step 2: Draw a diagram Draw a diagram to represent the situation. The rain is falling at an angle of 30 degrees to the vertical. The man is running horizontally at a speed of 10 m/s.
Step 3: Calculate the speed of the rain with respect to the man When the man is running, he perceives the rain to be falling vertically. This means that the horizontal component of the rain's velocity must be equal to the man's running speed. We can use trigonometry to find the horizontal component of the rain's velocity.
The horizontal component of the rain's velocity (Vx) can be calculated as follows: Vx = V * sin(30) = 10 m/s Where V is the speed of the rain.
Step 4: Calculate the speed of the rain with respect to the earth The speed of the rain with respect to the earth is the resultant of the horizontal and vertical components of the rain's velocity. The vertical component of the rain's velocity (Vy) can be calculated as follows: Vy = V * cos(30)
The speed of the rain with respect to the earth (V) can be calculated using the Pythagorean theorem: V = sqrt(Vx^2 + Vy^2)
By substituting the values we have: V = sqrt((10 m/s)^2 + (V * cos(30))^2)
Solving this equation will give the speed of the rain with respect to the earth.
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