A boat in a current The water in a river moves south at 10...

A boat in a current The water in a river moves south at $10 \mathrm{mi} / \mathrm{hr}$. If a motorboat is traveling 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

Relative Motion

Relative motion describes how the velocity of an object is perceived differently depending on the observer's frame of reference. In problems involving moving mediums, such as water or air currents, the object's velocity relative to the medium is found by accounting for the velocity of the medium itself.

Vector Addition and Subtraction

The process of combining or separating velocities in different frames is achieved through vector addition or subtraction. For example, to find the velocity relative to a moving medium, the medium's velocity vector is subtracted from the object's velocity vector relative to the shore.

Vector Components and Decomposition

Velocities are often broken down into perpendicular components (such as north-south and east-west) to simplify calculations. Decomposing vectors into components allows for straightforward application of algebraic addition or subtraction, which is essential in determining the resultant vector.

Magnitude and Direction Calculation

Once the resultant vector's components are determined, its overall speed (magnitude) is calculated using the Pythagorean theorem. The direction of the vector is then determined using trigonometric functions like the inverse tangent, providing a complete description of the motion in terms of both speed and angle.

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