Question

A ship leaves the port of Miami with a bearing of \(\mathrm{S} 80^{\circ} \mathrm{E}\) and a speed of \(15 \mathrm{knots}\). After 1 hour, the ship turns \(90^{\circ}\) toward the south. After 2 hours, maintaining the same speed, what is the bearing to the ship from the port?

Short answer

The ship's bearing from the port is approximately S 9.7°W.

Step-by-step solution

- Understand the initial movement

The ship leaves the port of Miami with a bearing of \(\text{S }80^{\circ} \text{E}\) and moves at a speed of 15 knots for 1 hour. This means the ship travels 15 nautical miles in this direction.

- Convert the bearing to a standard angle measurement

A bearing of \(\text{S }80^{\circ} \text{E}\) means the ship is traveling \((180^{\circ} - 80^{\circ}) = 100^{\circ}\) from the north, measured clockwise.

- Calculate the coordinates after the first leg of the journey

Using basic trigonometry, determine the displacement: \[x_1 = 15 \cos(100^{\circ}), \ y_1 = 15 \sin(100^{\circ}) \] \[ x_1 \approx -2.6 \text{ nautical miles}, \ y_1 \approx 14.8 \text{ nautical miles} \]

- Understand the second movement

After the initial hour, the ship turns 90 degrees toward the south and maintains the same speed for 2 hours, covering a distance of 2 \( \times \) 15 = 30 nautical miles directly south.

- Calculate the final coordinates

Add the second leg of the journey to the first: \[x_2 = x_1, \ y_2 = y_1 - 30 \] \[ x_2 \approx -2.6 \text{ nautical miles}, \ y_2 \approx 14.8 - 30 = -15.2 \text{ nautical miles} \]

- Determine the bearing from the port

Use the final coordinates to find the bearing: \[ \theta = \text{tan}^{-1} \left( \frac{|x_2|}{|y_2|}\right) = \text{tan}^{-1}\left( \frac{2.6}{15.2}\right)\] \ \[ \theta \approx 9.7^{\circ} \] \[ \text{Thus, the bearing is } 180^{\circ} + 9.7^{\circ} = 189.7^{\circ}, \text{ or } \text{S 9.7}^{\text{deg}}\text{W} \]

Key concepts

bearing calculation

Understanding how a ship's direction is described using bearings is essential in navigational trigonometry. Bearings are used to specify direction based on a 360-degree compass. This differs from the angular measurement in mathematics. Here, a bearing of \(\text{S }80^{\text{deg}} \text{E}\) indicates the ship is sailing 80 degrees east of south. To convert a bearing to a standard angle, measure clockwise from the north. Thus, \(\text{S }80^{\text{deg}} \text{E}\) translates to 100 degrees clockwise from north. Accurate bearing calculations help in plotting a ship's course, avoiding navigational errors.

trigonometric functions

Knowing trigonometric functions is crucial for determining a ship's displacement. Trigonometry utilizes functions like sine and cosine to calculate distances and angle-based coordinates. Given a bearing angle, cosine is used for the x-coordinate or east/west displacement, and sine is used for the y-coordinate or north/south displacement.
Using the bearing \( \text{S }80^{\text{deg}} \text{E} \), the ship's initial coordinates after one hour are:

\ x = 15 \text{cos}(100^{\text{deg}}) \
\ y = 15 \text{sin}(100^{\text{deg}}) \

On calculating:
\ x \ (approx. -2.6 nautical miles)
\ y \ (approx. 14.8 nautical miles).
These functions are powerful tools in solving navigational problems.

nautical miles displacement

Nautical miles serve as the unit of distance in maritime and air navigation. It's essential when measuring a ship's displacement over a set period. One nautical mile equals approximately 1.1508 miles or 1.852 kilometers.
In this exercise, the ship's speed is noted as 15 knots (nautical miles per hour), aiding in calculating its displacement after each leg of the journey. After one hour, it covers 15 nautical miles at a bearing of \(\text{S }80^{\text{deg}} \text{E}\). The ship then turns south and travels directly 30 nautical miles in two hours. Understanding nautical miles helps assess how far a vessel has moved on its course.

angle measurement conversion

When navigating, converting angle measurements between different formats is vital. Bearings are typically given in degrees with respect to compass directions. To use them in trigonometric calculations, convert these bearings to standard angular measurements.
For instance, in the initial leg, \(\text{S }80^{\text{deg}} \text{E}\) converts to:
\(180^{\text{deg}} - 80^{\text{deg}} = 100^{\text{deg}} \).
This standardized angle allows for consistent trigonometric function application.
After finalizing the displacement coordinates, converting angles aids in determiningthe ship's final bearing:
\( \text{tan}^{-1}\big(\frac{|x_2|}{|y_2|}\big) \).
Thus, bearings are simplified for practical navigation.