# Solving For Unknown Vector Components

1. Feb 1, 2010

1. The problem statement, all variables and given/known data

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?

2. Relevant 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°

3. The attempt at a solution

Last edited: Feb 1, 2010
2. Feb 1, 2010

### 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.

3. Feb 1, 2010

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

4. Feb 1, 2010

### rl.bhat

The two equations become
(Pi)x + (W)x = (Pf) ..........(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)

5. Feb 2, 2010

Thanks rl. Lets see if I am following you correctly.

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

(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°

Last edited: Feb 2, 2010
6. Feb 2, 2010

### 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?

7. Feb 2, 2010

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.

8. Feb 2, 2010

### 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.

9. Feb 2, 2010