# Obscure vector problem

• jemerlia

## Homework Statement

To a cyclist X traveling at 8 km h-1 due east the wind appears to
blow from the south. Another cyclist Y travels at 16 km h-1 at
N30°W relative to X. The direction of the wind as experienced by Y
is from the west. Calculate the speed and direction of the wind.

## Homework Equations

Vectors - relative motion

## The Attempt at a Solution

The problem appears slightly ambiguous in that I would assume that N30W is an absolute direction. A vector diagram might be drawn:
.......WY
......^--------->
....../
...XG..../YX
^----------->/
|
|WX
|

where
WX is wind velocity relative to X
XG is X velocity relative to ground
YX is Y velocity relative to X
WY is wind velocity relative to Y

The vectors could be added WG = WY+YX+XG
but, of course, no magnitude is given for WY and WX - only the direction.

I can't see how to proceed from here - help gratefully received :)

I wonder if taking Y's direction as an absolute value affects the problem - however I feel that if a compass bearing is quoted then it should be treated as a compass bearing - still puzzled by the word relative in "relative to X".

First of all you want to draw the cyclists as vectors traveling on their compass headings. X is easy.

X = 8 i + 0 j

However Biker Y is given relative to Biker X. This means that you need to account for Biker X's speed in determining Biker Y's true direction.

Y = - (16*Sin30° - 8) i + 16*Cos30° j

Then you need to figure what it means for Biker X to experience the wind as from the S and biker Y to experience it from the West.

Thanks for the hint. What is true is that WG (wind speed relative to the ground) must be the same for both cyclists. Following your suggestion I drew separate vector diagrams for the two cyclists:

...^.^
...WG.../...|
.../...|WX WG = WX +XG
.../.....|
------------->
XG

WY
^----------->
..\.../
.YG.\.../WG WG = WY+YG
...\../

It appears that the system might generate simultaneous equations allowing solution of j and i - but although I have given myself a 1h crash course in imaginary numbers I'm unsure of the next step and its mechanism - help gratefully received :)

I think that what you can conclude is that since each perceives only West or South and they are at right angles to each other, the Bicyclist 1 motion cancels the X component and perceives just the Y component (magnitude unknown) of the wind and Bicyclist 2's motion cancels the Y component of the wind and perceives only X directed wind. (I renamed them to avoid confusion.)

Hence taking the canceling components from their motions relative to the ground, the Wind is directed as = 8 i + 16*cos30 j

You can find |wind| through ordinary means.

Please forgive my unfamiliarity with imaginary numbers. Do the i's and j's also represent the y and x co-ordinates? I am unsure how to solve the expression to find the co-ordinates.

Please forgive my unfamiliarity with imaginary numbers. Do the i's and j's also represent the y and x co-ordinates? I am unsure how to solve the expression to find the co-ordinates.

So sorry. i,j's are simply vector notation for the vector components in the x,y directions respectively. If I could have written x-hat and y-hat more conveniently to emphasize the vector property I would have. It's just a short hand, and has nothing at all to do with imaginary numbers.

Once again - thank you. I'm still puzzled as to how solve for two unknowns in the expression for the wind direction...

the Wind is directed as = 8 i + 16*cos30 j

After further thought it was clear that your explanation was only a trivial step short of the answer. However, I would be grateful if you would please check that my reasoning is correct:

(a) Wind direction - the expression 8x + 16*cos30y (x,y instead of i,j) points to the co-ordinates (8, 16*cos 30). This yields an angle of 60degrees which corresponds with the given answer of blowing towards N30E.
(b) |wind| = SQRT(8^2 + (16*cos30)^2) = 16km/h - which again corresponds with the given answer.

Once again - thank you. I'm still puzzled as to how solve for two unknowns in the expression for the wind direction...

the Wind is directed as = 8 i + 16*cos30 j

While you do have 2 unknowns - namely the x-component and y component of wind - you can solve them independently by the fortuitous circumstance that the wind as perceived by 1 is only in one dimension at right angle to his motion (and at that he's cycling along an axis) and the wind perceived by 2 is at right angle to that.

Orthogonal is a wonderful thing.