Pilot Reem Ahmad's flight plan has her leaving Dulles International Airport and flying a Boeing 727 due east at 450 mph. There is a 65 mph wind with the bearing 60 degrees.
a. Determine the compass heading that Reem should follow.
b. Determine the airplane's ground speed. (Assume the speed with no wind is 450 mph)
Wind+heading= 450East
so whatever the NS compononent of wind must be counteracted by heading.
See if you can take if from here.
It's not making sense to me. Could you please explain further?
To determine the answers to both questions, we need to understand how the wind affects an aircraft's heading and ground speed using vector addition. Let's break down the steps:
a. The compass heading that Reem should follow:
To find the compass heading, we need to find the direction in which the aircraft should be pointing (relative to north) to counteract the wind and maintain a desired track. Start by drawing a diagram:
-----> (Direction of Flight - 450 mph)
|
Wind ---|
|
Since we have the wind speed and bearing, we can add this to the diagram:
-----> (Direction of Flight - 450 mph)
|\
Wind ---| \
|__\ 60°
Now, we need to find the resulting heading (compass direction) that the aircraft needs to follow. To do this, we use trigonometry. We can break down the wind vector into its x and y components with respect to the heading of the aircraft. Let's call the unknown component "x" and the known component "y":
x = Wind speed * cos(Wind bearing)
y = Wind speed * sin(Wind bearing)
Using the given values:
x = 65 mph * cos(60°)
y = 65 mph * sin(60°)
Now let's calculate these values:
x ≈ 32.5 mph
y ≈ 56.22 mph (rounded to 2 decimal places)
The x component represents the wind's effect on the aircraft's heading. Since the wind is coming from the northeast, we need to subtract this value from the desired heading (east) to compensate for it:
Resultant Heading = 90° - arctan(y/x)
Resultant Heading ≈ 90° - arctan(56.22/32.5)
Resultant Heading ≈ 90° - arctan(1.73)
Resultant Heading ≈ 90° - 59.04°
Resultant Heading ≈ 30.96°
Therefore, Reem should follow a compass heading of approximately 30.96°.
b. The airplane's ground speed:
The ground speed is the speed of the aircraft relative to the ground. We need to calculate this by finding the resultant of the aircraft's airspeed and the wind vector.
The ground speed can be found using the Pythagorean theorem:
Ground Speed^2 = (Airspeed)^2 + (Wind Speed)^2 + 2(Airspeed)(Wind Speed) * cos(Angle between them)
Given values:
Airspeed = 450 mph
Wind Speed = 65 mph
Angle between them = 180° - 60° = 120° (opposite direction)
Substituting the values into the formula:
Ground Speed^2 = (450 mph)^2 + (65 mph)^2 + 2(450 mph)(65 mph) * cos(120°)
Working through the calculation:
Ground Speed^2 = 202,500 + 4,225 - (58,500 * -0.5)
Ground Speed^2 = 206,725 + 29,250
Ground Speed^2 = 235,975
Taking the square root of both sides:
Ground Speed ≈ √235,975
Ground Speed ≈ 486.77 mph (rounded to 2 decimal places)
Therefore, Reem's airplane will have a ground speed of approximately 486.77 mph.