Suppose a pilot flies 40.0 km in a direction 60^∘ north of east and then flies 30.0 km in a di

step 1 In this example, we have a plane that goes along two straight paths. Initially, it goes a distan

Suppose a pilot flies 40.0 km in a direction 60^∘ north of...

Key Concepts

Vector Decomposition

This involves breaking a vector into its horizontal and vertical components using trigonometric functions. In navigation, decomposing directional movements into east-west and north-south components helps in analyzing overall displacement and understanding contributions from each directional leg.

Vector Addition

Vector addition is the process of combining vectors to determine a resultant vector. In the context of navigation, the pilot’s different legs of flight and any wind effects are combined by adding their components, which then yield the total displacement and direction from the starting point.

Trigonometry in Navigation

Trigonometry is essential in relating angles and sides in right triangles which are formed when decomposing vectors into components. It is used to compute the magnitude and direction of the resultant vector from these components, key in plotting a flight path or understanding changes due to wind.

Relative Velocity

Relative velocity examines the velocity of an object in relation to a moving medium—in this case, the airplane's speed relative to the surrounding air and the additional influence of wind. Understanding relative velocity is crucial for adjusting flight paths and ensuring accurate navigation.

Wind Correction

Wind correction involves adjusting an aircraft's heading to compensate for wind drift. The effect of wind on the flight path is dependent on both its speed and direction relative to the aircraft's air speed, and proper wind correction is essential to maintain the intended course.

Transcript

00:03 In this example, we have a plane that goes along two straight paths.

00:08 Initially, it goes a distance of 40 kilometers at an angle 60 degrees north of east.

00:16 We draw our compass here, northeast, southwest.

00:25 And then after that, it goes a distance of 30 kilometers 15 degrees north of east.

00:32 So that's what i've drawn here with the first part of our path being a and the second part being b.

00:38 So the resulting displacement vector, which we're going to call r in green here, is going to be the sum of a and b.

00:47 And we want to figure out what the magnitude of r is and also what angle it makes with the horizontal in our coordinate system here.

01:00 Okay.

01:01 So to do this, we're going to have to figure out what the components of r are.

01:05 And if we remember, when we add two vectors together, we can get the resulting vector just by adding the components of each vector together.

01:16 So that means we can get r sub x just by adding the x components of a and b.

01:22 And we can get r sub y by adding the y components of a and b.

01:28 So we just need to go work out what these components are.

01:31 And then we can look at calculating the magnitude and the direction of our vector.

01:36 So for a, let's start there.

01:42 We're going to find the x and y components by using our trigonometric functions sign and cosine.

01:50 Specifically, we have a right triangle formed by a that looks something like this.

02:02 And we're going to be able to get the y component of a over here by using the sign function because the sign relates the opposite and hypotenuse, or the opposite side in the hypotenuse of this triangle to the angle.

02:20 And we'll be able to get the adjacent side with the cosine, because the cosine is going to relate the adjacent side and the hypotenuse to the angle.

02:29 So a sub x is going to be given by a, the magnitude of our vector, or the hypotenuse of this triangle, times the cosine of 60 degrees.

02:44 And a sub y will be given by a times the sign of 60 degrees.

02:53 So for a sub x, this is just going to come out to 20 meters.

02:57 And for a sub y, we should get 34 .6 meters.

03:04 And that's just when we're using a is equal to 40 kilometers.

03:07 So actually, we should make sure we put kilometers here.

03:14 So let's get rid of those, put kilometers.

03:23 Now, for the x and y components of b, we're going to do something identical.

03:26 So again, our vector b, we can form a right triangle with it and our axes in our coordinate system here.

03:41 And we'll be able to get the adjacent side to our angle with the cosine and the opposite side with the sign.

03:50 So bx is going to be b times the cosine of 15 this time.

04:00 And b sub y is going to be b times the sign of 15 degrees.

04:05 So we should put 15 degrees here too.

04:09 Okay.

04:11 And if we go ahead and plug those into our calculator, where b is 30 kilometers, we should get 29 meters for b sub x and 7 .7 meters, set it again.

04:28 Kilometers, kilometers, 29 kilometers and 7 .77 kilometers for b sub y...

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