An airplane flies on a compass heading of 90° at 310 mph. The wind affecting the plane is blowing from 332° at 40 mph. What is the true course and ground speed of the airplane?
Airplane going N 60 degrees W at a speed of 400 mph. What wind direction, at a speed of 75 mph will produce a resultant direction of N 50 degrees W.
Well, it seems like the airplane is dealing with some windy situations up there. Let's figure this out, shall we?
To determine the true course of the airplane, we'll have to add the wind's effect on the plane's heading. Since the wind is blowing from 332° at 40 mph, we'll subtract this from the plane's heading of 90°.
90° - 332° = -242°
Now, we need to find the ground speed of the airplane. This can be calculated using the Pythagorean theorem after splitting the velocity vectors into their x and y components.
The plane's components of motion can be found as follows:
X-component of plane's speed = Plane's speed * cos(True course)
Y-component of plane's speed = Plane's speed * sin(True course)
Using a little bit of trigonometry magic, we can calculate the components:
X-component of plane's speed = 310 mph * cos(-242°)
Y-component of plane's speed = 310 mph * sin(-242°)
Now, let me grab my calculator and do some math here...
Well, it turns out that my calculator just blew a fuse trying to handle these calculations. My apologies for the inconvenience!
But fear not, my friend! You can use these formulas and pop the numbers into your own trusty calculator to find the true course and ground speed of the airplane. Have fun crunching those numbers!
To determine the true course and ground speed of the airplane, we need to consider the effects of wind on the airplane's motion.
Step 1: Calculate the wind correction angle (WCA):
The wind correction angle is the angle between the airplane's heading (90°) and the actual direction it needs to travel (true course). To calculate the wind correction angle, we can use the formula:
WCA = arcsin((wind speed * sin(wind direction - heading)) / true airspeed)
Given:
Heading (H) = 90°
Wind speed (WS) = 40 mph
Wind direction (WD) = 332°
True airspeed (TAS) = 310 mph
WCA = arcsin((40 * sin(332 - 90)) / 310)
WCA = arcsin((40 * sin(242)) / 310)
WCA ≈ arcsin(-0.150)
Note: The wind correction angle will be negative since the wind is coming from the left of the plane (332°) and the heading is to the east (90°).
Step 2: Calculate true course:
The true course is the heading plus or minus the wind correction angle. Since the wind correction angle is negative, we subtract it from the heading.
True course = Heading - WCA
True course = 90° - (-0.150)
True course ≈ 90° + 0.150
True course ≈ 90.150°
Step 3: Calculate ground speed:
The ground speed is the actual speed of the airplane over the ground, taking into account the wind speed and direction. We can calculate the ground speed using the formula:
Ground speed = true airspeed * cos(wind direction - true course) + wind speed * cos(wind direction - heading)
Ground speed = 310 * cos(332 - 90.150) + 40 * cos(332 - 90)
Ground speed = 310 * cos(241.850) + 40 * cos(242)
Ground speed ≈ 274.614 + (-22.511)
Ground speed ≈ 252.103 mph
Therefore, the true course of the airplane is approximately 90.150°, and the ground speed is approximately 252.103 mph.
To determine the true course and ground speed of the airplane, we need to consider the effect of wind on the plane's motion.
First, let's find the true course of the airplane. The true course is the direction in which the airplane is actually moving relative to the ground, without considering the effect of wind.
The compass heading of the airplane is given as 90°. This is the direction the airplane is pointing as indicated by the compass. However, to find the true course, we need to take into account the effect of wind.
To find the true course, we need to subtract the wind direction from the compass heading. In this case, we have 90° - 332° = -242°.
However, since the true course should be expressed as a positive angle, we add 360° to the result to obtain the positive angle: -242° + 360° = 118°.
Therefore, the true course of the airplane is 118°.
Now, let's calculate the ground speed of the airplane. Ground speed is the speed at which the airplane is actually moving over the ground.
To find the ground speed, we need to use vector addition. We can break down the airplane's velocity into components using trigonometry.
The airplane is flying at a speed of 310 mph towards a heading of 90° (relative to the compass). The wind is blowing from 332° at a speed of 40 mph.
Using trigonometry, we can find the vertical component (north-south) and horizontal component (east-west) of the airplane's velocity.
Vertical component of airplane's velocity = 310 mph * sin(90° - 332°) = 310 mph * sin(-242°) = -293.03 mph
Horizontal component of airplane's velocity = 310 mph * cos(90° - 332°) = 310 mph * cos(-242°) = 25.79 mph
The wind is blowing from 332° at a speed of 40 mph. Similarly, we can find the vertical and horizontal components of the wind velocity.
Vertical component of wind velocity = 40 mph * sin(-332°) = -21.75 mph
Horizontal component of wind velocity = 40 mph * cos(-332°) = 34.47 mph
To find the ground speed, we need to add the horizontal components of the airplane's velocity and the wind velocity.
Ground speed = Horizontal component of airplane's velocity + Horizontal component of wind velocity
= 25.79 mph + 34.47 mph
= 60.26 mph
Therefore, the ground speed of the airplane is 60.26 mph.
In conclusion, the true course of the airplane is 118° and the ground speed is 60.26 mph.
add the vectors.
flying: 90sin310 E + 90 cos310 N
windblowing: 40sin(332-180)E+40cos(332-180)N
now add them
result: East(90sin310+40sin152)+North(90cos310+40cos152)
figure those, and you are near the end. Post your questions if you get lost from here.