A steamer is moving due east with 36 km / h. To a man in the steamer the wind appears to blow a

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A steamer is moving due east with 36 km / h. To a man in...

A steamer is moving due east with $36 \mathrm{~km} / \mathrm{h}$. To a man in the steamer the wind appears to blow at $18 \mathrm{~km} / \mathrm{h}$ due north. Find the velocity of the wind. (A) $5 \sqrt{5} \mathrm{~ms}^{-1} \tan ^{-1} \frac{1}{2}$ North of East (B) $5 \mathrm{~ms}^{-1} \tan ^{-1} 2$ North of East (C) $5 \sqrt{5} \mathrm{~ms}^{-1} \tan ^{-1} 2$ North of East (D) $5 \mathrm{~ms}^{-1} \tan ^{-1} \frac{1}{2}$ North of East

A steamer is moving due east with $36 \mathrm{~km} / \mathrm{h}$. To a man in the steamer the wind appears to blow at $18 \mathrm{~km} / \mathrm{h}$ due north. Find the velocity of the wind.
(A) $5 \sqrt{5} \mathrm{~ms}^{-1} \tan ^{-1} \frac{1}{2}$ North of East
(B) $5 \mathrm{~ms}^{-1} \tan ^{-1} 2$ North of East
(C) $5 \sqrt{5} \mathrm{~ms}^{-1} \tan ^{-1} 2$ North of East
(D) $5 \mathrm{~ms}^{-1} \tan ^{-1} \frac{1}{2}$ North of East

Key Concepts

Trigonometric Analysis for Magnitude and Direction

After determining the components of a vector, the overall magnitude and direction can be found using the Pythagorean theorem and inverse trigonometric functions. This process is key to converting from component form back to a magnitude-angle representation, which describes the speed and the angle of the actual wind.

Coordinate Systems and Component Resolution

Establishing a coordinate system—such as east and north directions—allows one to decompose velocity vectors into horizontal and vertical components. This decomposition simplifies the process of vector addition or subtraction and is crucial for analyzing motions in different directions.

Relative Velocity

Relative velocity is the concept that the speed and direction of an object can appear different when measured from a moving reference frame. In problems like these, the apparent wind experienced by an observer on the moving steamer is the difference between the actual wind velocity and the velocity of the steamer itself.

Vector Addition and Subtraction

This concept involves representing physical quantities such as velocity as vectors which have both magnitude and direction. To find the true wind velocity, one subtracts the steamer's velocity vector (the observer's velocity) from the apparent wind vector, essentially performing vector subtraction.

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