A man running on a horizontal road at 8 ms^-1 finds rain falling vertically. If he increases hi

step 1 In this question we will learn about some basic concept of the vectors. The question is a man ru

A man running on a horizontal road at 8 ms^-1 finds rain...

A man running on a horizontal road at $8 \mathrm{~ms}^{-1}$ finds rain falling vertically. If he increases his speed to $12 \mathrm{~ms}^{-1}$, he finds that drops make $30^{\circ}$ angle with the vertical. Find velocity of rain with respect to the road. (A) $4 \sqrt{7} \mathrm{~ms}^{-1}$ (B) $8 \sqrt{2} \mathrm{~ms}^{-1}$ (C) $7 \sqrt{3} \mathrm{~ms}^{-1}$ (D) $8 \mathrm{~ms}^{-1}$

A man running on a horizontal road at $8 \mathrm{~ms}^{-1}$ finds rain falling vertically. If he increases his speed to $12 \mathrm{~ms}^{-1}$, he finds that drops make $30^{\circ}$ angle with the vertical. Find velocity of rain with respect to the road.
(A) $4 \sqrt{7} \mathrm{~ms}^{-1}$
(B) $8 \sqrt{2} \mathrm{~ms}^{-1}$
(C) $7 \sqrt{3} \mathrm{~ms}^{-1}$
(D) $8 \mathrm{~ms}^{-1}$

Key Concepts

Relative Velocity

This concept refers to the velocity of an object as observed from a particular frame of reference. In problems like this, the apparent direction of the falling rain changes with the observer’s motion, and by comparing the observed velocities on different frames, one can relate the rain’s true velocity to the observer’s speed.

Vector Decomposition

This involves breaking down a vector into its horizontal and vertical components. By resolving the rain's velocity into these components, one can use the information about the angle it makes with the vertical to relate the horizontal and vertical speeds using trigonometric ratios.

Reference Frame Transformation

This is the concept of analyzing the motion from different perspectives—typically the stationary ground frame and the moving observer’s frame. The changes in the observed direction of the rain when the observer alters his speed provide insight into the actual horizontal velocity component of the rain in the ground frame.

Trigonometric Analysis of Vectors

Trigonometry is used to connect the angle at which the rain appears to fall with its velocity components. By relating the tangent of the deviation angle to the ratio of the horizontal component of the rain’s velocity (relative to the observer) to its vertical component, one can establish equations that ultimately yield the rain’s true velocity.

Transcript

Please give Ace some feedback