Problem 49 - Precalculus : Building Concepts and Connections

Question

Round your answers to two decimal places. The world's largest weathervane is located in Montague, Michigan. On a July day in 2007 , it showed that the wind had a speed of 15 miles per hour in the direction \(S 30^{\circ} \mathrm{E}\). Express the wind velocity in component form. (Source: www. wunderground.com)

Step-by-Step Solution

Answer

The wind's velocity in component form is thus approximately \(v = (-7.5 mph, -12.99 mph)\).

Step 1: Breaking down the compass direction

The compass direction given is normally expressed as degrees moving clockwise from North. However, here we have \(S 30^{\circ} \mathrm{E}\), which implies that from South, the wind is moving 30 degrees towards the East. Position the compass directions in a standard Cartesian coordinate plane in your mind: North (N) is up (along positive y-axis), East (E) is right (along positive x-axis), South (S) is down (negative y-axis) and West (W) is to the left (negative x-axis). Now, this has to be translated into the angular measurement from the positive x-axis typically used in trigonometry calculations.

Step 2: Translate the compass direction into angular measurement

The direction from South increases towards East, which in a standard Cartesian coordinate plane goes in the negative direction of x. Therefore, the angular measurement becomes \(270^{\circ} - 30^{\circ} = 240^{\circ}\).

Step 3: Express the wind's velocity in terms of its x and y components

In a polar coordinate system, the x (i.e., horizontal or east-west) and y (i.e., vertical or north-south) components of a vector can be determined using the vector's magnitude (in this case, speed) and its direction (angle). The x component is the product of the magnitude and the cosine of the angle, whereas the y component is the product of the magnitude and the sine of the angle. Here, because of the angle direction already discussed, the x-component formula is \(v_x = |v| * cos(\theta)\)and the y-component formula is \(v_y = |v| * sin(\theta)\), where |v| is the magnitude of the vector (wind speed) and \( \theta \) is the angle.

Step 4: Calculate the x and y components

Substitute the given speed value (15 mph) and the calculated angle (240 degrees) into the formulas to get \(v_x = 15 * cos(240^{\circ})\) which results in \(v_x = -7.5 mph\) (rounded to 2 decimal places) and \(v_y = 15 * sin(240^{\circ}) \) which results in \(v_y = -12.99 mph\) (rounded to 2 decimal places).

Key Concepts

Trigonometry in Physics

Trigonometry, the study of triangle angles and sides, plays an integral role in physics, particularly when breaking down vectors into their components. In the context of measuring wind velocity, trigonometry allows us to use the magnitude and direction of the wind to find the speed at which it moves horizontally and vertically.

Using sine and cosine functions from trigonometry, we can convert the speed and direction information into a mathematical representation. For instance, the cosine function relates the angle and the horizontal (x) component, and the sine function relates the angle and the vertical (y) component of a vector. By applying these functions, we can calculate precisely how much of the wind's speed is going in each direction, no matter the angle—essential for predicting weather patterns and planning in aviation and sailing.

Polar Coordinate System

Unlike the Cartesian coordinate system which uses a grid of x and y values, the polar coordinate system defines a location with just two values: a radius and an angle. In this system, the radius indicates how far away a point is from the origin, while the angle shows the direction from a fixed line, generally the positive x-axis.

In our weathervane example, the wind's speed can be considered the radius (the distance from the origin), and the given direction (e.g. 'S 30° E') can be translated into an angle in the polar coordinate system. This is immensely useful as it simplifies the representation of wind direction, which naturally fits a circular, rather than a rectangular, pattern.

Vector Components

Any vector, such as wind velocity, can be broken down into component parts, typically along the x (horizontal) and y (vertical) axes. This is crucial for analyzing forces, velocities, and other directional quantities in physical problems.

The horizontal component of a vector tells us how much of the vector is going in the east-west direction, while the vertical component tells us the north-south direction. For instance, a wind blowing from the south towards the east has both eastward (x) and southward (y) components. By finding these components, it becomes easier to understand and utilize the vector in calculations and practical applications, like plotting an aircraft's trajectory against the wind.

Compass Direction Angles

Compass directions, like 'S 30° E', can be somewhat ambiguous and need to be converted into a more standard form that can be used in calculations. These direction angles define the angle of a vector in relation to a fixed direction, usually north or east.

In order to use trigonometry, these angles must be translated into the standard mathematical angle measurements, which typically take counterclockwise rotation from the positive x-axis as positive angles. Once translated, these angles serve as the basis for breaking down vectors into their x and y components, allowing us to navigate, plot, and predict using mathematical precision.