How Do Planes Calculate Relative Velocities?

[takingphysics]

Homework Statement

Two airplanes taxi as they approach the terminal. Plane 1 taxies with a speed of 13.4 m/s due north. Plane 2 taxies with a speed of 6.2 m/s in a direction 18.8° north of west.
(a) What are the direction and magnitude of the velocity of plane 1 relative to plane 2?
(b) What are the direction and magnitude of the velocity of plane 2 relative to plane 1?

Homework Equations

The law of cosines: A^2 = B^2 + C^2 - 2*B*C*cos(A)

The Attempt at a Solution

I set the vectors tail - to - tail and used the law of cosines to solve for the resultant vector, which I call n:

n^2 = 6.2^2 + 13.4^2 - 2(6.2)(13.4)cos(71.2)
n = 12.823 m/s

So I have the magnitude. Now I need the direction. For (a), the direction will be X degrees north of east, and for (b) the direction will be Y degrees south of west. I'm have a feeling that X=Y, but I can't seem to get the right answer. Here's what I'm doing:

I solve for the angle between the 12.823 and the 18.8 sides:
theta = cos^-1 ([12.823^2 + 18.8^2 - 13.4^2]/[2*12.823*18.8])
theta = 45.4385 degrees

That leaves the remaining angle in the triangle to be 63.3615 degrees.

So to find the direction for, let's say, part (b), shouldn't I just subtract 64.3615 degrees from 90 degrees? That would yield 25.6385 degrees. But that answer is incorrect.

What am I failing to see?

Thanks for your response!

 

[takingphysics]

At the very end of my post I meant to say 63.3615 and 26.6385 degrees; it was a typo.

 

[baba89]

I would like to point out that the given information is not sufficient to accurately determine the direction and magnitude of the velocity of one plane relative to the other. We would need additional information such as the initial positions and directions of the planes, as well as the time elapsed for their movement, in order to accurately calculate their relative velocities.

However, based on the given information, your approach using the law of cosines to find the resultant velocity is correct. For part (a), the direction of the velocity of plane 1 relative to plane 2 would be 45.4385 degrees north of east, and for part (b), the direction of the velocity of plane 2 relative to plane 1 would be 134.3615 degrees south of west.

It is possible that the answer given for part (b) is incorrect, as the direction should be in the range of -180 degrees to +180 degrees. I would recommend double checking your calculations and using a calculator or software that can handle complex vector operations.

Additionally, as a scientist, I would also suggest including units in your calculations and clearly labeling the final answers to avoid any confusion. I hope this helps and good luck with your homework!