The water in a river moves south at 10 mi / hr . A motorboat travels due east at a speed of 20 m

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The water in a river moves south at 10 mi / hr . A motorboat...

The water in a river moves south at $10 \mathrm{mi} / \mathrm{hr} .$ A motorboat travels due east at a speed of
$20 \mathrm{mi} / \mathrm{hr}$ relative to the shore. Determine the speed and direction of the boat relative to the moving water.

Key Concepts

Magnitude and Direction Calculation

Once a resultant vector is obtained through vector subtraction, calculating its magnitude involves using the Pythagorean theorem, and finding its direction relative to a reference axis involves using trigonometric functions such as the arctangent. This process converts vector components into a measurable speed and an angle that describes the direction of the resultant vector.

Vector Subtraction

Vector subtraction is used to compute the relative velocity between entities moving in different frames. By subtracting the vector representing the medium's velocity from the object’s velocity vector, one obtains the effective velocity relative to the moving medium.

Relative Velocity

Relative velocity is the velocity of one object as observed from a reference frame moving with another object. In problems like this, velocities must be adjusted by subtracting the velocity of the moving medium from the object's velocity measured relative to a fixed frame, ensuring that the correct frame of reference is used to describe motion.

Vector Representation and Components

In analyzing motion, velocity is represented as a vector with both magnitude and direction. Breaking down a velocity vector into perpendicular components (such as east-west and north-south) simplifies calculations by allowing separate treatment of each directional component in subsequent operations.

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