vector problem...?

Question

An airplane is flying at 150 mi/hr (its speed in still air) in a direction such that a wind of 60 mi/hr blowing east to west, the airplane travels in a straight line southward....

what must the plane's heading direction for it to fly diredtly south?...

how do you do this?... i thought that i could add the Vector of the plane relative to the earth with the Vector of the wind relative to the earth to get the resultant, answer... but don't i need at least one angle to do that? or am i over thinking it and there is an easier way to figure this out... please help
thank you...

Answer 1

You're told the plane is flying southward because of the wind so in effect, you are givne the resultant vector. The problem is asking you to find the plane's vector such that when added to the wind, the plane flies south.

Let south be 180 degrees from north, east 90 degrees from north, west 270.

Now the wind is blowing at vw at an angle of 270 deg

the plane is moving at vp, at and angle of 180 deg

Now by addition: v0^2 =vw^2+vp^2 where v0 is plane's speed without the wind.

So v0 = 161.5 mph

Since the plane flies due south, the wind velocity must equal the planes speed in teh opposite direction. This speed is:

ve = v0*cos(q)= vw where q = angle between plane's vector and east.


so q = arccos(vw/v0) = 68.2 deg south of east or 185.2 degrees from north

Answer 2

The plane will be heading SE, at an angle such that its airspeed (150 mph) is the hypotenuse of a right triangle, with the wind speed (60 mph) one of the triangle's legs. The angle is arcsin(60/150) east of due south.