Problem 54 - College Trigonometry

Question

The HMS Sasquatch leaves port with bearing \(\mathrm{S} 20^{\circ} \mathrm{E}\) maintaining a speed of 42 miles per hour (that is, with respect to the water). If the ocean current is 5 miles per hour with a bearing of \(\mathrm{N} 60^{\circ} \mathrm{E}\), find the HMS Sasquatch's true speed and bearing. Round the speed to the nearest mile per hour and express the heading as a bearing, rounded to the nearest tenth of a degree.

Step-by-Step Solution

Answer

The true speed of the HMS Sasquatch is 42 mph, with a bearing of S 17.6° E.

Step 1: Convert Bearings to Angles

First, convert the bearings to angles measured counterclockwise from the positive x-axis, which is east. For the ship's bearing, `S 20° E` translates to an angle of 200° (180° + 20°) from the x-axis. For the ocean current's bearing, `N 60° E`, the angle is 60° from the positive x-axis.

Step 2: Resolve Velocities into Components

Resolve the velocities of the ship and the ocean current into their respective x (eastward) and y (northward) components. The ship's velocity components are calculated as: - Eastward component: 42 mph * sin(200°) - Northward component: 42 mph * cos(200°) For the ocean current, the components are: - Eastward component: 5 mph * cos(60°) - Northward component: 5 mph * sin(60°)

Step 3: Calculate True Velocity Components

Add the respective components of velocity from the ship and the ocean current to get the resultant velocity components (true speed). - True eastward component = ship's eastward component + current's eastward component - True northward component = ship's northward component + current's northward component.

Step 4: Calculate True Speed

Determine the resultant speed using the Pythagorean theorem on the true velocity components. \[\text{True speed} = \sqrt{(\text{True eastward component})^2 + (\text{True northward component})^2}\]Round this speed to the nearest mile per hour.

Step 5: Determine True Bearing

Find the angle for the true bearing using the inverse tangent function:\[\theta = \tan^{-1}\left(\frac{\text{True northward component}}{\text{True eastward component}}\right)\]Adjust \(\theta\) to match the compass bearing format, considering in which quadrant the resultant vector lies. Convert \(\theta\) into a bearing, rounded to the nearest tenth of a degree.

Key Concepts

Vector Addition

Vector addition is a fundamental concept in trigonometry and physics, used here to determine the resultant velocity of the HMS Sasquatch. It involves combining two or more vectors to get a single resultant vector, which represents the combined effect of the individual vectors. This is crucial when dealing with forces or velocities that are not aligned with one another.
To add vectors, like in our problem, we break each vector into components. In our case, we resolve the ship's and the current's velocities into eastward and northward components.

• Vertical (north-south) components contribute to the combined northward direction.
• Horizontal (east-west) components contribute to the combined eastward direction.

By adding these components, we get a new set of resultant components. This process helps in visualizing how different forces or actions, like a ship moving in water with current, influence the final path or velocity.

Bearing Calculation

Bearing calculation is essential for navigation and involves determining the direction of travel relative to the north direction. In the original exercise, we are required to determine the true bearing of the HMS Sasquatch after accounting for the ocean current.
A bearing is expressed in degrees, using three figures, often incorporating cardinal directions (N, E, S, W). For instance, `S 20° E` bearing leads to a path south, turned 20 degrees eastward.
To perform a bearing calculation after resolving velocities into components, we determine the resultant vector's angle using the inverse tangent function.
This angle then needs adjustment to standard compass terms, considering:

• The quadrant in which the vector lies.
• Converting mathematical angles to compass bearings.

Practicing bearing calculations sharpens navigation skills, crucial for tasks like steering a ship.

Polar Coordinates

Polar coordinates are a way of plotting or describing points in a plane where each point is determined by a distance from a reference point and an angle from a reference direction. Unlike Cartesian coordinates which use x and y-axes, polar coordinates are more intuitive for angles and distances.
They are particularly useful in navigation, especially for describing movements and bearings, as they align with the circular nature of compass bearings and distance measurement.

• The radial coordinate represents the vector's magnitude or speed.
• The angular coordinate represents direction, comparable to bearing.

In our exercise, after resolving the ship and ocean current velocities, we can view these components similarly to polar coordinates, using magnitude and direction to understand their combined influence. This allows us to view navigation problems through both coordinate systems, offering flexibility in solution approaches.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used to calculate the resultant speed in vector addition problems. It states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): \[c^2 = a^2 + b^2\].
In vector addition, this becomes crucial when determining the magnitude of a resultant vector from its components.
For the HMS Sasquatch, once the true velocity components (eastward and northward) are found, the Pythagorean theorem calculates the overall true speed. By squaring each component, summing them, and taking the square root, we find the true speed:\[\text{True speed} = \sqrt{(\text{True eastward})^2 + (\text{True northward})^2}\].
This method is reliable and simple, showing the power of basic geometry in solving complex real-world navigation challenges.