Finding the Correct Compass Heading The pilot of an aircraft...

Finding the Correct Compass Heading The pilot of an aircraft wishes to head directly east but is faced with a wind speed of $40 \mathrm{mph}$ from the northwest. If the pilot maintains an airspeed of $250 \mathrm{mph}$, what compass heading should be maintained to head directly east? What is the actual speed of the aircraft?

Key Concepts

Compass Heading

Compass heading is a term used to describe the direction in which an aircraft is pointed relative to the cardinal points, typically measured in degrees. Understanding compass heading is key in navigation as it ensures the aircraft is directed along a specific course, even when compensating for external factors like wind.

Relative Velocity

Relative velocity in the context of navigation refers to the actual speed and direction of an aircraft over the ground. It is obtained by combining the aircraft's airspeed with the wind's velocity, represented as vectors. This concept is essential for determining both the groundspeed and the required heading adjustments.

Trigonometric Decomposition

Trigonometric decomposition involves breaking down a vector into its horizontal and vertical components using sine and cosine functions. This method is crucial when analyzing the effects of wind on the aircraft, as it allows pilots to resolve the wind’s impact along and perpendicular to the intended path.

Vector Addition

Vector addition is a fundamental concept in navigation where the different velocity vectors, such as an aircraft's airspeed vector and the wind's velocity vector, are combined to determine the resultant vector, known as the groundspeed. This process enables the pilot to understand how external forces like wind affect the actual path of the aircraft.

Wind Correction Angle

The wind correction angle is the adjustment made to an aircraft’s heading to counteract the effect of wind drift. By calculating this angle, a pilot can steer the aircraft in a manner that neutralizes the crosswind component, ensuring that the desired ground track, such as flying directly east, is maintained.

Transcript

00:01 So we know that the airplane is traveling at 250 miles per hour and wants to end up going east.

00:06 However, there's a wind that is going northwest at 40 miles per hour.

00:11 So if we draw a little picture down here and let's start with the wind.

00:15 So here's a north, south, east, west.

00:21 And there's our vector for the wind.

00:24 And this is a 45 degree and 45 degree angle.

00:26 And we know that that angle ends up being 135 degrees in here.

00:32 So this angle is 135 degrees.

00:35 Now, the resultant vector is this plane does want to end up going east.

00:44 And so let's do another north, south, east, west right here.

00:49 And that airplane is going to end up having to divert its, and this magnitude is 40.

00:57 So we'll kind of do that.

00:58 That should be about roughly six times longer.

01:01 So that plane is going to need to fly at an angle that is going to get it where this magnitude is 250.

01:14 And this little angle that's right in here, this theta, would actually be equivalent to this theta because this is a east -west line and this is an east -west line.

01:28 So we actually can use law of signs.

01:31 And the law of signs would be that the sign of that little angle theta is to its side opposite, which is 40, as the sign of this angle of 135 degrees is to its side opposite.

01:50 And so the sign of theta, and we move that up here, we have 40 times the sign of 135, which we already know is square root of 2 over 2...

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