The pilot of an aircraft wishes to fly due
west in a 52.6 km/h wind blowing toward the south. The speed of the aircraft in the absence of a wind is 154 km/h.
A.) How many degrees from west should the aircraft head? Let clockwise be positive. Answer in units of ◦ (degrees)
B.) What should the plane’s speed be relative to the ground? Answer in units of km/h.
X = -154 km/h.
Y = -52.6 km/h = Speed of wind.
A. tanAr = Y/X = -52.6/-154 = 0.34156.
Ar = 19o = Reference angle.
A = Ar + 180 = 19 + 180 = 199o. = 19o
South of West.
The plane should head 19o North of West.
B. V^2 = X^2 + Y^2.
V^2 = (-154)^2 + (-52.6)^2 = 26482.76
V = 163 km/h.
To answer these questions, we need to break down the components of the aircraft's motion and the wind.
Let's start with the first question:
A.) How many degrees from west should the aircraft head?
To determine the heading of the aircraft, we need to find the angle between its motion and the due west direction. This angle is formed due to the combined effect of the aircraft's speed and the wind.
To calculate this angle, we can use the concept of a vector diagram. Let's draw one:
- Draw a horizontal line to represent the west direction.
- Draw another line representing the aircraft's actual velocity direction (in the absence of wind) relative to the ground. This line would be pointing towards the left (west). We'll call this line Vaircraft.
- Draw a vertical line representing the wind's velocity. This line would be pointing towards the south. We'll call this line Vwind.
Now, draw a line connecting the end points of Vaircraft and Vwind. This line represents the resulting velocity vector (Vresultant) of the aircraft's velocity relative to the ground.
Now that we have this vector diagram, we need to find the angle between Vaircraft and Vresultant. We can do this by using trigonometry.
Given:
- Speed of the aircraft in the absence of wind = 154 km/h
- Wind speed = 52.6 km/h
- The wind is blowing towards the south
First, calculate the magnitude of Vresultant using the Pythagorean theorem:
Magnitude of Vresultant = sqrt((154 km/h)^2 + (52.6 km/h)^2)
Next, calculate the angle using trigonometry:
Angle = tan^(-1)(52.6 km/h / 154 km/h)
This calculation gives us the angle between the aircraft's heading and the due west direction. But for the desired answer, we need to measure the angle clockwise from the west direction. We can achieve this by subtracting the calculated angle from 90 degrees.
B.) What should the plane’s speed be relative to the ground?
To calculate the speed of the aircraft relative to the ground, we use the magnitude of Vresultant obtained in the previous question.
The speed of the aircraft relative to the ground is the magnitude of Vresultant.
Therefore, the plane's speed relative to the ground is equal to the magnitude of Vresultant obtained in the previous question.