The Gulf Stream off the east coast of the United States can...

The Gulf Stream off the east coast of the United States can flow at a rapid $3.6 \mathrm{m} / \mathrm{s}$ to the north. A ship in this current has a cruising speed of $10 \mathrm{m} / \mathrm{s}$. The captain would like to reach land at a point due west from the current position. a. In what direction with respect to the water should the ship sail? b. At this heading, what is the ship's speed with respect to land?

The Gulf Stream off the east coast of the United States can flow at a rapid $3.6 \mathrm{m} / \mathrm{s}$ to the north. A ship in this current has a cruising speed of $10 \mathrm{m} / \mathrm{s}$. The captain would like to reach land at a point due west from the current position.
a. In what direction with respect to the water should the ship sail?
b. At this heading, what is the ship's speed with respect to land?

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Key Concepts

Relative Velocity

Relative velocity is the concept of how the velocity of an object is observed in different frames of reference. In navigation, it is crucial to account for the movement of the medium (such as a current or wind) relative to the ground, and then combine it with the vehicle’s own velocity through the medium to determine its overall path.

Vector Decomposition and Addition

This concept involves breaking down a velocity into its perpendicular components (for example, northward and eastward components) using trigonometric relationships. The resultant velocity or overall path of an object moving in a medium is the sum of its own velocity vector and the velocity vector of the medium, calculated through vector addition.

Trigonometry in Navigation

Trigonometry is used to calculate the correct heading by relating the components of vector quantities through functions like sine and cosine. It helps in determining the necessary angle of deviation from a reference direction so that the effect of an external influence, such as a current, is offset, ensuring that the resultant path meets the intended course.

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