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Chapter 12: Problem 226

A man walks at \(5 \mathrm{~km} / \mathrm{h}\) in the direction of a \(20 \mathrm{~km} / \mathrm{h}\) wind. If raindrops fall vertically at \(7 \mathrm{~km} / \mathrm{h}\) in still air, determine direction in which the drops appear to fall with respect to the man.

Short Answer

Expert verified
With respect to the man, the raindrops appear to fall towards him, against his direction of movement.

Step by step solution

01

Determine the Velocity of the Man Relative to the Ground

The man is walking at \(5 \, \text{km/h}\). This is his velocity relative to the ground. Let's denote it as \(\vec{v}_m\).
02

Determine the Velocity of the Wind Relative to the Ground

The wind speed is \(20 \, \text{km/h}\). This is its velocity relative to the ground. Let's denote it as \(\vec{v}_w\).
03

Determine the Velocity of the Raindrops in Still Air

The velocity of the raindrops in still air is \(7 \, \text{km/h}\), falling vertically. This can be denoted as \(\vec{v}_r\).
04

Calculate the Relative Velocity of the Man with Respect to the Wind

The velocity of person with respect to the wind is the velocity of the man subtracted from the velocity of the wind. Thus, \(\vec{v}_{mw} = \vec{v}_m - \vec{v}_w = 5 \, \text{km/h} - 20 \, \text{km/h} = -15 \, \text{km/h}\). The negative sign indicates that the man is moving against the direction of the wind.
05

Calculate the Relative Velocity of Raindrops With Respect to the Man

The velocity of the raindrops with respect to the man is the sum of the velocity of the raindrops and the velocity of the man relative to the wind. So, \(\vec{v}_{rm} = \vec{v}_r + \(\vec{v}_{mw} = 7 \, \text{km/h} - 15 \, \text{km/h} = -8 \, \text{km/h}\). The negative sign indicates that, to the man, the raindrops appear to be coming at an angle against his direction of movement.

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