A bird is flying from Hamilton ON to Waterloo ON. There is a heavy wind traveling at 5.0km/h (S11°E). What should its heading be? How long will it take?
That is all the information I get and I am confused as to how to approach this problem because I don't have another velocity. I did some research and to get to Hamilton to Waterloo is 68km at a heading of (N60°W). I don't know if this information is relevant or even needed to answer the problem. Thank you in advance.
There is always a useful right-angled triangle in a problem of this kind; but it does not necessarily have any sides parallel to either of your coordinate axes.
I prefer to represent the wind velocity as a vector pointing in the same direction that the particles of air are moving. So if the direction the wind is coming from is $11^\circ$ east of due south, the direction it is going to is $11^\circ$ west of due north.
The line the bird wants to travel (its course) is $60^\circ$ west of due north. The angle between this and the wind direction is $49^\circ$. So the first right triangle to consider is one with an angle of $49^\circ$ adjacent to the course from Hamilton to Waterloo and a hypotenuse of $5$ km/h.
In order for the bird to fly in the directin of Waterloo in this wind, its velocity vector plus the wind's velocity vector must be a vector directly toward Waterloo. SO draw a vector from the tip of the wind's vector back to the course the bird wants to fly. Now you have a second right triangle with one leg on the course, sharing its other leg with the other triangle.
Notice that the hypotenuse of the new triangle must be at least as large as the shared leg. If the bird flies slower than $5 \sin 49^\circ$ km/h, it will not even be able to stay on course.
To fully solve the problem you need the bird's speed relative to the surrounding air (its true airspeed). Without this information, you can set up equations and find a formula that gives the combined speeds of bird and wind along the course to Waterloo, from which you can determine the time taken, as well as the angles of the second triangle, from which you can determine the bird's heading. These formulas are the best answer you can give to a question with this amount of information, I think.