An airplane pilot wishes to maintain a true course in the direction 250° with a ground speed of 400 mi/hr when the wind is blowing directly north at 60 mi/hr. Approximate the required airspeed and compass heading
To approximate the required airspeed and compass heading, you need to consider the effect of the wind on the true course. Here's how you can proceed:
1. Draw a diagram: Draw a diagram representing the situation. Label the true course direction of 250° and indicate the north wind blowing at 60 mi/hr. This will help you visualize the relationship between the true course and the resulting course.
2. Calculate the resulting course: To determine the resulting course, you need to subtract the wind correction angle from the true course. The wind correction angle can be calculated using the following formula:
Wind Correction Angle = arcsin(wind speed/ground speed) [in radians]
In this case, wind speed = 60 mi/hr and ground speed = 400 mi/hr. Convert mi/hr to radians/hr by dividing by the radius of the Earth in statute miles (approximately 3,963 mi).
Wind Correction Angle = arcsin(60/400) = arcsin(0.15) ≈ 8.59°
Subtracting the wind correction angle from the true course gives:
Resulting Course = True Course - Wind Correction Angle
Resulting Course = 250° - 8.59° ≈ 241.41°
3. Calculate the required airspeed: To calculate the required airspeed, you need to determine the component of the wind along the resulting course. This can be done using the following formula:
Component of Wind = Wind Speed * cos(Resulting Course - True Course)
In this case, wind speed = 60 mi/hr and the resulting course = 241.41°.
Component of Wind = 60 * cos(241.41° - 250°)
Component of Wind ≈ 60 * cos(-8.59°) ≈ 53.675 mi/hr (round to the nearest hundredth)
The required airspeed is the sum of the ground speed and the component of the wind:
Required Airspeed = Ground Speed + Component of Wind
Required Airspeed = 400 + 53.675 ≈ 453.68 mi/hr (round to the nearest hundredth)
4. Calculate the compass heading: Lastly, to find the compass heading, you need to compensate for the drift caused by the crosswind. The drift angle can be calculated using the following formula:
Drift Angle = arcsin(crosswind speed/ground speed) [in radians]
In this case, the crosswind speed is the wind speed perpendicular to the resulting course:
Crosswind Speed = Wind Speed * sin(Resulting Course - True Course)
Crosswind Speed = 60 * sin(241.41° - 250°)
Crosswind Speed ≈ 60 * sin(-8.59°) ≈ -13.42 mi/hr (round to the nearest hundredth)
The compass heading can be obtained by adding the drift angle to the resulting course:
Compass Heading = Resulting Course + Drift Angle
Compass Heading ≈ 241.41° + arcsin(-13.42/400)
Compass Heading ≈ 241.41° + arcsin(-0.03355) ≈ 241.41° - 1.92° ≈ 239.49° (round to the nearest hundredth)
Therefore, the required airspeed is approximately 453.68 mi/hr and the compass heading is approximately 239.49°.
To maintain a true course in the direction 250°, the airplane pilot needs to consider the wind effect and adjust the airspeed and compass heading accordingly. Here are the step-by-step calculations:
Step 1: Calculate the wind correction angle (WCA):
The wind correction angle is the angle between the direction of the aircraft's heading and the direction of the aircraft's track over the ground.
WCA = arctan(wind speed / true airspeed)
Given:
Wind speed = 60 mi/hr
True airspeed = unknown (let's call it TAS)
WCA = arctan(60 / TAS)
Step 2: Calculate the true heading:
The true heading is the compass heading corrected for the wind effect.
True heading = True course + WCA
Given:
True course = 250° (desired direction)
True heading = 250° + WCA
Step 3: Calculate the groundspeed component due to the crosswind:
The crosswind component is the component of the wind that acts perpendicular to the aircraft's heading.
Crosswind component = wind speed * sin(WCA)
Crosswind component = 60 * sin(WCA)
Step 4: Calculate the airspeed required to maintain the desired groundspeed:
The airspeed required to maintain the desired groundspeed is the sum of the groundspeed and the crosswind component.
Required airspeed = groundspeed + crosswind component
Required airspeed = 400 + 60 * sin(WCA)
Combining all the steps, the approximate required airspeed and compass heading can be determined using the given information. The exact values will depend on the actual wind correction angle calculated in Step 1:
Required airspeed ≈ 400 + 60 * sin(arctan(60 / TAS))
True heading ≈ 250° + arctan(60 / TAS)
I have a triangle with sides 60 and 400 and the contained angle is 70°
let the resultant be R
R^2= 60^2 + 400^2 - 2(60)(400) cos70°
= 147183.0331
R = 383.64 mi/h
finding the angle opposite the 70°
sinØ/60 = sin70/R
sinØ = .14696..
Ø = 8.45°
so the compass heading is 250 + 8.45 = 258.5°