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A man is travelling at 10.8 kmph in a to...

A man is travelling at 10.8 kmph in a topless car on a rainy day. He holds an umbrella at an angle of 37° with the vertical so that he does not wet. If rain drops falls vertically downwards, what is the rain velocity ?

A

`1 ms^(-1)`

B

`2 ms^(-1)`

C

`3 ms^(-1)`

D

`4 ms^(-1)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine the velocity of the rain (V_rain) given the velocity of the car (V_car) and the angle at which the umbrella is held (θ). Here’s a step-by-step breakdown of the solution: ### Step 1: Understand the Problem The car is moving horizontally at a speed of 10.8 km/h, and the rain is falling vertically downwards. The umbrella is held at an angle of 37° with the vertical to prevent the man from getting wet. ### Step 2: Convert the Speed of the Car Convert the speed of the car from km/h to m/s for consistency in units. \[ V_{car} = 10.8 \text{ km/h} = \frac{10.8 \times 1000 \text{ m}}{3600 \text{ s}} = 3 \text{ m/s} \] ### Step 3: Draw a Diagram Visualize the situation: - The velocity of the car (V_car) is horizontal. - The velocity of the rain (V_rain) is vertical. - The umbrella makes an angle of 37° with the vertical. ### Step 4: Determine the Angles Since the umbrella is at an angle of 37° with the vertical, the angle with the horizontal (θ) is: \[ θ = 90° - 37° = 53° \] ### Step 5: Use Trigonometry Using the tangent function, which relates the opposite side (V_rain) to the adjacent side (V_car): \[ \tan(θ) = \frac{V_{rain}}{V_{car}} \] Substituting the known values: \[ \tan(53°) = \frac{V_{rain}}{3 \text{ m/s}} \] ### Step 6: Calculate the Tangent Calculate the value of \(\tan(53°)\): \[ \tan(53°) \approx 1.327 \] ### Step 7: Solve for V_rain Now, substituting this value into the equation: \[ 1.327 = \frac{V_{rain}}{3} \] Thus, \[ V_{rain} = 1.327 \times 3 \text{ m/s} \approx 3.981 \text{ m/s} \] ### Step 8: Round the Result Rounding this to a more manageable figure, we can say: \[ V_{rain} \approx 4 \text{ m/s} \] ### Final Answer The velocity of the rain is approximately **4 m/s**. ---
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