Vector Operations - Resultant Ground Speed and Direction of Plane

[TheUppercut]

Homework Statement

An airplane is traveling with airspeed of 225 mph at a bearing of 205 degrees. A 60 mph is blowing with a bearing of 100 degrees. What is the resultant ground speed and direction of the plane?

Homework Equations

x = u cos(degrees)
y = v sin(degrees)

However, I think the solution needs to be found by using the Law of Cosines and the Law of Sines.

Law of Cosines: c^2 = a^2 + b^2 - (2)(a)(b)(cos(C))

Law of Sines: (sin(a)/A) = (sin(b)/B)

The Attempt at a Solution

I was able to find the solution by using the first method, but not by using the Law of Cosines.

First method:
x = 225 cos(205) + 60 cos(100) = -214.338 --> -214.338^2 = 45940.839
y = 225 sin(205) + 60 sin (100) = -36 --> -36^2 = 1296

x^2 + y^2 = 45940.839 + 1296 = 47236.839 --> sqrt(47236.839) = 217, which is the ground speed

I then found the bearing by taking the tan(y/x), then arctan of that answer. So, (-36/-214.338) = .16796 --> arctan(.16796) = 9.53, which has to be added to 180, to get the bearing of 189.53

With the Law of Cosines and Law of Sines, I am completely stumped. I have tried drawing a diagram with the bearings and then using the Law of Cosines to find the resultant, but no such luck. Any help would be greatly appreciated!

 

[tiny-tim]

Welcome to PF!

Hi TheUppercut! Welcome to PF!

(try using the X² icon just above the Reply box )

With the Law of Cosines and Law of Sines, I am completely stumped. I have tried drawing a diagram with the bearings and then using the Law of Cosines to find the resultant, but no such luck.

You have two sides, and the angle between them (85°), so the cosine rule should work …

Show us your full calculations, and then we'll see what went wrong!

 

[TheUppercut]

Still getting acclimated to the whole thing, thank you for your patience!

I don't understand where you are getting 85 from.

But if I were to use 85, it would be something like:

c²=225² + 60² - (2)(225)(60)(cos(85)
c² = 51871.75
c = sqrt (51871.75) = 227 (** the answer is 217)

And now because that angle is wrong, the Law of Sines would not work either. I think the angle has to be around 75, but again, I'm not sure where you would come up with that number. I've tried subtracting (360-205) to get 155, and then (180-100) to get 80, then 155-80 = 75, but there is no way that can be right.

In my diagram, my 205 degree bearing is in the 3rd quadrant, and the 100 degree bearing is in the 2nd quadrant. Maybe the problem cannot be solved with the Law of Cosines and Law of Sines, but the example in the book had a similar problem that was solved with both Laws.

Again, I really appreciate everyone's help and insights!

 

[tiny-tim]

oops!

I don't understand where you are getting 85 from.

oops! nor do I!

Should be 75° (adding the vectors head-to tail, of course).

Sorry!

 

[TheUppercut]

Forgive me for being dense, but how would head-to-tail work in this instance? Would you take the non-arrow end of the 100 degree bearing and place it on the arrow end of the 205 degree bearing? But even then, how would that give you 75?

From there, you would still need at least one more angle to be able to use the Law of Sines, and I'm absolutely stumped as to how to find them.

I appreciate your patience.

 

[tiny-tim]

Hi TheUppercut!

… how would head-to-tail work in this instance? Would you take the non-arrow end of the 100 degree bearing and place it on the arrow end of the 205 degree bearing? But even then, how would that give you 75?

You always add vectors head-to-tail (if you do it head to head, you get the difference, not the sum).

From there, you would still need at least one more angle to be able to use the Law of Sines, and I'm absolutely stumped as to how to find them.

uhh?

There's only 3 angles in a triangle …

if you know 2, then you know the third anyway, since they add to 180° …

the sine rule is intended to work with only 1 known angle!

 

[TheUppercut]

I guess I didn't phrase my question correctly. I'm just unable to see where the addition of a 205 degree bearing with a 100 degree bearing gives you 75 degree angle to use.

I'm just having a lot of trouble visualizing the addition of the vectors on a diagram, and then seeing the angles they create...if that wasn't already obvious.

(Those FNGs could screw up anything, couldn't they?)

 

[TheUppercut]

I have no idea why I made it seem like I needed all of the angles to use the Law of Sines; that was stupid of me.

I still don't see where you were able to come up with the 75 degree angle from the addition of a 205 degree bearing and a 100 degree bearing.

 

[tiny-tim]

I still don't see where you were able to come up with the 75 degree angle from the addition of a 205 degree bearing and a 100 degree bearing.

Ah, look at the diagram …

although you're adding the vectors, to get the angle between them, you have to subtract the angles, don't you?

 

[TheUppercut]

Was I right in doing that then in post #3 above?
Or are you going about it a different way?

 

[tiny-tim]

I'm confused …

which part of post #3 are you asking about? ​

 

[TheUppercut]

I'm so sorry for being difficult

360 - 205 = 155
180 - 100 = 80
155 - 80 = 75

I think I figured out how to do the head-to-tail addition. But the actual addition of the bearings (205 and 100) is still confusing me as to how to get 75. I'm baffled

I'm just not seeing it. Once we get that, then we can get the Law of Sines to work. It's not like I don't know how to solve the problem the one way; I just want to be well-versed in both methods.

 

[tiny-tim]

But the actual addition of the bearings (205 and 100) is still confusing me as to how to get 75. I'm baffled

No, you don't add them …

the angle between requires you to subtract …

205° minus 100° = 105° (and it's the complement of that, ie 75°, for reasons difficult to explain but which should be obvious when you look at the diagram ).

 

[TheUppercut]

Awesome, thank you! I didn't even think of the complement!

Now,

(sin 75)/(217) = (sin A) / 225

(sin 75) * (225) / 217 (which is actually 217.34) = .999966

arcsin (.999966) = 89.53

When I use the arctan in my first method, I knew to add 180 because both my y and x were negative. Here, though, is different. What tells us to add 100 to 89.53 to get the final answer?

 

[tiny-tim]

(now I come to think of it again, it's the supplement, not the complement!)

What tells us to add 100 to 89.53 to get the final answer?

It's usually safest to look at the diagram.

In this case, the angle is so close to 90° that looking at the diagram doesn't really help, so I'd have used the sine rule to calculate the other angle instead.

 

[TheUppercut]

Complement (90), Supplement (180)...I knew what you meant!

So, when we use the Law of Sines to get the other angle, we get 15.49

(sin 75 * 60) / 217 = .267
arc sin (.267) = 15.49

205 - 15.49 = 189.51

Or is there a better way to do that? That still doesn't seem right.

 

[tiny-tim]

So, when we use the Law of Sines to get the other angle, we get 15.49

i'd do it like this …

so the bearing is 15.49° to the right of 25° down and left …

that's 9.51° down and left, or an absolute bearing of 189.51°

 

[TheUppercut]

Thank you so much for your help and for being so welcoming to the site.