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Question

Rain is falling vertically downwards with a speed of 30 m/s. A man is running towards east with a speed of 10 m/s. At what angle with the vertical should he hold his umbrella to protect himself from the rain?

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [Option B]
## CONCEPT FRAMEWORK
	**Primary Concept**: Relative Velocity
	**Related Topics**: Vector Addition, Trigonometry
	**Prerequisites**: Basic understanding of vectors and trigonometric functions.
## SOLUTION PATHWAY
**Given Information**:
	* Velocity of rain ($v_r$) = 30 m/s (vertically downwards)
	* Velocity of man ($v_m$) = 10 m/s (towards east)
	* We need to find the angle at which the man should hold the umbrella to protect himself from rain.
**Method & Steps**:
	1. Determine the relative velocity of rain with respect to the man.
		Reasoning: The man perceives the rain falling at a velocity which is the vector difference between the rain's velocity and his own velocity. To protect himself, he should hold his umbrella parallel to this relative velocity.
		Formula/Rule: $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m$
	2. Visualize the vectors: The rain's velocity is downwards, and the man's velocity is towards the east.  The relative velocity of rain with respect to the man will be the vector difference, which can be visualized by placing the tail of the man's velocity at the head of the rain's velocity. The resultant vector will point in the direction of the rain relative to the man.
		Working: $\vec{v}_{rm}$ will have a vertical component equal to the rain's velocity (30 m/s) and a horizontal component equal to the negative of the man's velocity (-10 m/s) in the east direction.
	3. Calculate the angle the relative velocity makes with the vertical.
		Working: Let $	heta$ be the angle the relative velocity makes with the vertical.  Then, $	an(	heta) = \frac{	ext{horizontal component}}{	ext{vertical component}} = \frac{10}{30} = \frac{1}{3}$.  Therefore, $	heta = \arctan(\frac{1}{3})$.
		Key Point: The umbrella should be held at the same angle as the relative velocity of the rain, but in the opposite direction.
	4.  Final Step:  The angle the man should hold the umbrella with the vertical is $\arctan(\frac{1}{3})$.
		Result: $	heta = \arctan(\frac{1}{3})$
		Verification: The angle calculated aligns with the direction of relative velocity.
## OPTION ANALYSIS
[A] :x: $\arctan(3)$ is the angle the relative velocity makes with the horizontal, not the vertical.
[B] ✓: $\arctan(\frac{1}{3})$ is the correct angle the relative velocity makes with the vertical.
[C] :x: $45^{\circ}$ is not the correct angle as the velocities are not equal.
[D] :x: $30^{\circ}$ is not the correct angle.
## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing the relative velocity with the actual velocity of rain or the man. [How to avoid]: Always consider the vector difference of velocities.
		2.[Error 2]: Taking the ratio of vertical velocity to horizontal velocity to calculate the angle. [How to avoid]: Remember to take the ratio of horizontal to vertical to find the angle with the vertical.
	**Quick Tips**:
		1.[Helpful shortcut]: Draw a vector diagram to visualize the relative velocity. This makes it easier to identify the correct trigonometric ratio.
		2.[Memory aid]: Remember that the angle of umbrella is with the vertical and the tangent is opposite over adjacent.

Mr. ANIL KUMAR.G · Answer

$\arctan(3)$

Mr. ANIL KUMAR.G · Answer

The correct Option is "$\arctan(\frac{1}{3})$"

Mr. ANIL KUMAR.G · Answer

$45°$

Mr. ANIL KUMAR.G · Answer

$30°$

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