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This article covers the steps involved in finding the wind and jet vector.

Following are the concepts covered and features of this article

  • How to interpret the bearing angles?
  • Steps to take the positive angle for the velocity vector
  • Finding the velocity vector of wind and jet plane.
  • Calculating the true sped of the jet plane
  • Finding the resultant direction of the jet plane.
  • Summary steps to find the true velocity of a jet plane.

Problem: A jet is flying through a wind that is blowing with a speed of 40mph in the direction N10E. The jet has a speed of 625mph relative to the air, and the pilot heads the jet in the direction N20W.

  1. Express the velocity of the wind as a vector in component form.
  2. Express the velocity of the jet relative to the air in component
  3. Find the true velocity of the jet as a vector.
  4. Find the true speed and direction of the Jet palne

 

Detailed Solution:-

 Finding the  velocity vector component form of wind:

Given that the wind is blowing with a speed (S1)=40mph and the direction is N 10 E.

So this representation conveys that, starting from the north direction, move the angle towards the East direction by 10 degrees(Shortest part is taken).

Therefore the terminal side of the angle makes  10 degrees with the north direction as shown in the picture.

 

 

Now we take the  Positive angle in degrees (anticlockwise direction), starting from the  East direction. Till the terminal side of the angle. Form the above picture, it is evident that the positive angle is  X=90-10 = 80 degrees.

Since we have the angle in degrees, we can find the Vector component form of wind using the formula

V1= S1 cos(X)*i+S1 sin(X)*j.

Hence  the vector component of wind is

V1= 40* cos(80)i+40* sin(80)j.

V1= (6.94592 )i+(39.39231)j

Therefore ,the V1= <6.94592,39.39231>

 

Finding the  velocity vector component of the JET Plane:

Given that the  Jet is flying with a speed (S2)=625mph and the direction is N 20 W.

Starting from the north direction, the angle is moved towards the west direction by 20 degrees.

As shown in the picture below the red angle in the clockwise direction is 20 degrees.

The picture below explains it.

Therefore Positive angle in degrees, starting from the  East direction. Till the terminal side of the angle, Y is 90+20=110 degrees.

Hence  the vector component of the jet is

V2= 625* cos(110)i+ 625* sin(1100)j.

V2= <-213.7625 ,587.30788>

 

Calculating the TRUE velocity  vector components of the Jet Plane

The effective velocity (V)of the Jetplane is  given by  V1 +V2

= <6.94592-213.7625 , 39.39231+587.30788>

V=<−206.8165, 626.7001>

Calculating the TRUE speed of the Jet Plane from TRUE Velocity Vector.

Speed is a scalar quantity. True speed is also called as ground speed.

Once we have the TRUE velocity vector, we can find the magnitude of the velocity vector.

lVl = \sqrt{\left( -206\cdot 185\right) ^{2}+\left( 626\cdot 700\right) ^{2}}

lVl =659.9439mph.

 

Finding the direction  of the Jet Plane:

The direction of the jetplane needs more attention and the concept of vectors play a key role.

Theta =\tan ^{-1}\left( -\dfrac{626.7001}{-206.8165}\right) = -71.7366.

Let us ignore the negative  angle that we  get using the formula

So this angle is 71.7336° and this is the angle from the “x” axis.

The green angle will be  90-71.7336=18.2664°.

Therefore the angle is 18.3°  rounded to one decimal place.

The direction in which the jet flies are N 18.3° W

OR

18.3° (West of North).

Steps to find the true jet  velocity :

Finding the vector component form of wind and the JET Plane.

  • Find the positive angle x in degrees(taken in anticlockwise direction from 0 degrees ) to the terminal side of the wind direction.
  • Similarly, Find the positive angle Y in degrees to the terminal side of the jet plane.
  • Find the velocity vector of the wind “W” with the given speed of the wind and the angle  X calculated.
  • Similarly, find the velocity vector of the JET  “W” with the given speed of the JET and the angle  Y calculated.

Finding the TRUE velocity  vector of the Jet Plane 

  • Add the wind and jet velocity vectors  to get the  resultant  True velocity

Finding the TRUE speed and direction of the Jet Plane.

  • Find the magnitude of the TRUE velocity vector and the direction using the arctan formula.