The Question is from neet-physics motion-in-a-straight-line , and it was created by the author NAGA SAI

Question

Rain is falling vertically downwards with a velocity $v_R$. A man walks with a velocity $v_M$. What is the angle with the vertical at which the man should hold his umbrella to protect himself from the rain?

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option C
## CONCEPT FRAMEWORK
	**Primary Concept**: Relative Velocity
	**Related Topics**: Vector Addition, Trigonometry
	**Prerequisites**: Understanding of vectors and their components, basic trigonometry.
## SOLUTION PATHWAY
**Given Information**:
	* Velocity of rain with respect to ground: $\vec{v_R}$ (vertically downwards)
	* Velocity of man with respect to ground: $\vec{v_M}$ (horizontally)
**Method & Steps**:
	1.  **Determine the relative velocity of rain with respect to the man**
		Reasoning: The man needs to hold the umbrella in the direction opposite to the relative velocity of rain with respect to himself. The relative velocity of rain with respect to the man is given by:  $\vec{v_{RM}} = \vec{v_R} - \vec{v_M}$
		Formula/Rule: $\vec{v_{RM}} = \vec{v_R} - \vec{v_M}$
	2.  **Visualize the vectors**
		Working: Since $\vec{v_R}$ is vertically downwards and $\vec{v_M}$ is horizontal,  $-\vec{v_M}$ will be in the opposite horizontal direction.  The resultant vector $\vec{v_{RM}}$ forms the hypotenuse of a right-angled triangle with sides representing $v_R$ (vertical) and $v_M$ (horizontal)
		Key Point: The relative velocity vector points in the direction of the apparent rain for the man.
	3. **Find the angle with the vertical**
		Result: The angle $	heta$ between the relative velocity vector and the vertical is given by $	an(	heta) = \frac{v_M}{v_R}$,  therefore, $	heta = 	an^{-1}(\frac{v_M}{v_R})$
		Verification: The derived formula matches option C.
 ## OPTION ANALYSIS
[A] :x: Incorrect. This represents the angle if the man's velocity was in the vertical direction and the rain was horizontal.
[B] :x: Incorrect. The sine function is used when dealing with the ratio of opposite side to hypotenuse which is not the case here.
[C] ✓: Correct. This gives the angle between the relative velocity vector and the vertical.
[D] :x: Incorrect. The cosine function is used when dealing with the ratio of adjacent side to hypotenuse which is not the case here.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. **Incorrectly subtracting velocities**:  When calculating relative velocity, make sure to subtract the velocity of the observer from the velocity of the object.
		2. **Mixing up the sides of the triangle**: When using trigonometry, make sure to correctly identify the adjacent and opposite sides with respect to the angle.
	**Quick Tips**:
		1. **Draw a vector diagram**: Visualizing the velocities as vectors can help in understanding the problem and arriving at the correct answer.
		2. **Think about what the man sees**: The umbrella should be held opposite to the direction the rain appears to be coming from for the man.

NAGA SAI · Answer

$	an^{-1}(\frac{v_R}{v_M})$

NAGA SAI · Answer

$\sin^{-1}(\frac{v_M}{v_R})$

NAGA SAI · Answer

The correct Option is "$	an^{-1}(\frac{v_M}{v_R})$"

NAGA SAI · Answer

$\cos^{-1}(\frac{v_M}{v_R})$

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