A girl standing on road holds her umbrella at $45^{\circ}$ with the vertical to keep the rain away. If she…

A girl standing on road holds her umbrella at $45^{\circ}$ with the vertical to keep the rain away. If she starts running without umbrella with a speed of $15\sqrt{2} \, \text{km} \, h^{-1}$, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is
  1. 30 km h-1
  2. 252 km h-1
  3. 302 km h-1
  4. 25 km h-1

Solution

v→rg=v→r-v→g

 

From the above diagram, velocity of the raindrop wrt ground will be,

v→r=1522+1522=30 km h-1

And velocity of the raindrop wrt the moving girl will be,

v→rg=152=302 km h-1

Alternative Solution: 1. Key Concepts: The key concepts involved in this problem are relative velocity and vector addition. In particular, we are dealing with the relative velocity of raindrops with respect to the girl. 2. Step-by-step Solution Process: Step 1: Analyze the given data. The angle at which the umbrella is held (45°) tells us the direction of the rain with respect to the ground. The girl is running at a speed of $15\sqrt{2}$ km/h. Step 2: Find the velocity of the rain with respect to the ground. The direction of rain with respect to the ground is at an angle of 45° with the vertical. So, the rain's velocity can be resolved into two components: one vertical (downwards) and the other horizontal (opposite to the girl's direction). If we assume the rain's speed to be V km/h, then both the horizontal and vertical components will be $V\cos(45°)$ = $V/\sqrt{2}$. Step 3: Find the relative velocity of the rain with respect to the girl. When the girl starts running, the horizontal component of the rain's velocity with respect to her becomes $V/\sqrt{2}$ + $15\sqrt{2}$ (since the rain's horizontal velocity and the girl's velocity are in opposite directions). Step 4: Determine the condition for the rain to hit the girl vertically. For the rain to hit the girl vertically, the horizontal component of the rain's velocity with respect to her must be zero. This means that $V/\sqrt{2}$ + $15\sqrt{2}$ = 0. Solving this equation gives $V = 30/\sqrt{2}$ km/h. Therefore, the correct answer is C) $\frac{30}{\sqrt{2}}$ km/h. 3. Explanation of the Correct Answer and Why Other Options are Wrong: The correct answer is C) $\frac{30}{\sqrt{2}}$ km/h because that is the speed at which the raindrops must be falling for them to hit the girl vertically when she is running at a speed of $15\sqrt{2}$ km/h. The other options are incorrect because they do not satisfy the condition for the rain to hit the girl vertically when she is running at the given speed. 4. Tips to Remember this Concept: Remember that when dealing with relative velocity problems, you need to take Alternative Solution: 1. Key Concepts Involved: The key concepts involved in this question are: a. Relative Velocity: It is the velocity of an object as observed from another moving object. b. Vector Addition: It is used to combine velocities in different directions. c. Trigonometry: It is used to resolve velocities into components. 2. Step-by-step solution process: Step 1: Understand the problem The problem says that when the girl stands still, the rain falls at an angle of 45° to the vertical. This means that the horizontal and vertical components of the rain's velocity are equal (since tan(45°) = 1). When the girl starts running at a speed of $15\sqrt{2}$ km/hr, the rain appears to be falling vertically on her. This means that the horizontal component of the rain's velocity is equal to the girl's running speed so that they cancel each other out. Step 2: Find the horizontal component of the velocity of rain The horizontal component of the rain's velocity is equal to the girl's running speed, which is $15\sqrt{2}$ km/hr. Step 3: Find the vertical component of the velocity of rain The vertical component of the rain's velocity is equal to the horizontal component, which is also $15\sqrt{2}$ km/hr. Step 4: Find the resultant velocity of rain with respect to the girl The resultant velocity of rain with respect to the girl is the vector sum of the horizontal and vertical components. As they form a right angle, we can use the Pythagorean theorem to find the magnitude of the resultant velocity, which is the hypotenuse of the right triangle formed by the horizontal and vertical components. So, the magnitude of the resultant velocity is $\sqrt{(15\sqrt{2})^2 + (15\sqrt{2})^2} = 15\sqrt{2}\sqrt{2} = 30\ km/hr$. Therefore, the speed of the rain drops with respect to the moving girl is $30\sqrt{2}\ km/hr$. Hence, the correct answer is (C) $30\sqrt{2}\ km/hr$. The other options are incorrect because they do not match the calculated speed of the rain drops with respect to the girl. Tips to remember this concept: Keep in mind that the angle at which rain falls can change depending on the observer's motion.

Asked in: JEE Main 2022 (27 Jun Shift 1)

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