Solving For Unknown Vector Components

[Badmachine]

Homework Statement

Is it possible to determine aircraft speed prior to contact with noted wind, if resulting aircraft speed, original aircraft heading, wind speed and wind direction and final aircraft heading are known?

Homework Equations

P (plane): Bearing 241° (traveling approximately southwest) @ "X" f/s.
W (wind): Traveling south @ 32 f/s.

Plane and wind components represented by ordered pairs:

P = ["X" f/s cos(241°), "X" f/s sin(241°)] = ?, ?
W = [-32 f/s cos(90°), -32 f/s sin(90°)] = 0, -32

? + 0 = ?
? + (-32) = ?

||P + W|| = ?² + ?² = ?^(1/2) = 710 f/s

tan −1(?/?) = 62.24° + 180° = 242.24°

Displacement = 242.24° - 241° = 1.24°

The Attempt at a Solution

 

[rl.bhat]

Select a co-ordinate axis so that final direction of the plane (Pf) is along -x-axis.
Let the angle of the initial direction of the plane (Pi) with he new x-axis be α.
Let the angle of the direction of the wind (W) with he new x-axis be β. From the given data you can find β.
Now sum of the x-components of Pi and W is equal to Pf.
And y component of Pi is equal and opposite to y-component of W.
Write the equations and solve.

 

[Badmachine]

Does the solution require eliminating the wind components or inverting the wind compinent's values?

 

[rl.bhat]

Does the solution require eliminating the wind components or inverting the wind compinent's values?

The two equations become
(Pi)_x + (W)_x = (P<sub>f) ...(1)

(Pi)y + (W)y = 0...(2)
Rearrange the equations.

(Pi)x = - (W)x + (Pf) ...(3)

(Pi)y = -(W)y ...(4)
Square eq. 3 and 4 and add to get (Pi)</sub>

 

[Badmachine]

Thanks rl. Let's see if I am following you correctly.

(Pi)x = - (W)x + (P<sub>f)

(Pi)y = -(W)y


(Pi)x = - (0)x + (P[710 f/s]f)

(Pi)y = -(32)y

710² + 32² = 505,124^(1/2) = 710.72

Apparently I'm performing the calculations incorrectly, as Pi = 683 f/s (value omitted in original post).

Original problem:

P (plane): Bearing 241° (traveling approximately southwest) @ 683 f/s (465 mph); W (wind): traveling south @ 32 f/s (22 mph)

[683 f/s cos(241°), 683 f/s sin(241°)] = -331, -597
[-32 f/s cos(90°), -32 f/s sin(90°)] = 0, -32

-331 + 0 = -331
-597 + (-32) = -629

||P + W|| = 331² + 629² = 505, 2021/2 = 710 f/s

tan −1(629/331) = 62.24° + 180° = 242.24°

Displacement = 242.24° - 241° = 1.24°</sub>

 

[rl.bhat]

What is required in the original problem?
For example, if the want to cross a river and reach the opposite bank, you have to row in the upstream direction. Similarly which velocity is given in the problem?

 

[Badmachine]

What is required in the original problem?

To determine the final aircraft velocity (Pf) and angular aircraft displacement from original heading, after application of a given wind.

Had hoped to learn how to "reverse engineer" for the original aircraft speed (Pi) or even heading simply based on other knowns.

 

[rl.bhat]

OK. It can be done in the following way. Refer my post#2 and #4
Pi*cosα + W*cosβ = Pf Or
Pi*cosα = Pf - W*cosβ
Pi*sinα = W*sinβ
Pi^2 = ( Pf - W*cosβ )^2 + (W*sinβ)^2
In the given problem, if the velocity of plane is Pf, then β = 270degrees - 241 degrees.
So using the above equation you can find Pi.

 

[Badmachine]

Thanks rl.

Perhaps I miscalculated somehwere, but at this stage I arrive at Pi = 649 f/s (as opposed to 683 f/s)

β = 270degrees - 241 degrees = 29
Pi^2 = ( Pf - W*cosβ )^2 + (W*sinβ)^2
Pi^2 = [710 - (-32)*cos29]^2 + [(-32)*sin29)]^2
421,400 = 421,159 + 241
649 f/s