A plane is flying at 168 miles per hour with a heading of 43.5° from due north. The wind is blowing a constant 33 miles per hour at 133.5° from due north. Find the ground speed and true course of the plane.
168 @ 43.5° = <115.6,121.9>
33 @ 133.5° = <23.94,-22.72>
add up the x- and y-components and the resultant velocity is
<139.54,99.18> = 171.20 @ 54.6°
45
To find the ground speed and true course of the plane, we need to decompose the velocities into their north and east components and then add them up.
Let's start by drawing a diagram to visualize the situation. We have the plane flying at a heading of 43.5° from due north, and the wind blowing at an angle of 133.5° from due north.
First, we need to decompose the plane's velocity into north and east components. To do that, we can use trigonometry. The north component of the plane's velocity can be calculated using the formula:
north component = velocity * sin(heading)
Substituting the values, we get:
north component = 168 * sin(43.5°)
north component ≈ 113.39 mph
Similarly, the east component of the plane's velocity can be calculated using the formula:
east component = velocity * cos(heading)
Substituting the values, we get:
east component = 168 * cos(43.5°)
east component ≈ 121.70 mph
Now, let's decompose the wind velocity into its north and east components. Using the same formulas, we can calculate:
north component (wind) = 33 * sin(133.5°)
north component (wind) ≈ -27.03 mph
east component (wind) = 33 * cos(133.5°)
east component (wind) ≈ -8.85 mph
Negative values indicate that the wind is blowing in opposite directions to our chosen positive directions.
To find the total north component, we add the north component of the plane's velocity to the north component of the wind velocity:
total north component = north component (plane) + north component (wind)
total north component ≈ 113.39 mph - 27.03 mph
total north component ≈ 86.36 mph
To find the total east component, we add the east component of the plane's velocity to the east component of the wind velocity:
total east component = east component (plane) + east component (wind)
total east component ≈ 121.70 mph - 8.85 mph
total east component ≈ 112.85 mph
Now that we have the north and east components of the total velocity, we can use these components to calculate the ground speed and true course.
The ground speed (GS) can be found using the Pythagorean theorem:
GS = √(north component^2 + east component^2)
Substituting the values, we get:
GS = √(86.36^2 + 112.85^2)
GS ≈ √(7452.1796 + 12714.5225)
GS ≈ √(20166.7021)
GS ≈ 142.02 mph
The true course (TC) can be found using the inverse tangent function:
TC = arctan(east component / north component)
Substituting the values, we get:
TC = arctan(112.85 / 86.36)
TC ≈ arctan(1.304)
TC ≈ 51.57° (rounded to two decimal places)
Therefore, the ground speed of the plane is approximately 142.02 mph, and the true course is approximately 51.57°.