I want to find the direction and how fast the plane travel relative to the ground? Physics!?

Question

A plane is to fly due noth, the speed of the plane relative to air is 200 km/h and the wind is blowing from west to east at 90 km/h. (a) in which direction should the plane head? (b) How was does the plane travel relative to the ground?

Answer 1

Let's assume that the speed through the air is TAS (true air speed) and not IAS (indicated air speed) and that the reference to the cardinal points is true north and not magnetic. (sorry but I am myself a pilot! :-)
You solve your resulting speed by Pythagoras: square root of (90exp2 + 200exp2) is 219.32 km/h.
Your heading will then be the arc tangent of 90 divided by 200 or, 24.7 degrees. If the wind is from the west, you will be heading true north but moving to a track of 24.7 degrees to the east, i.e. to the north-north-east.
Note an interesting question: If you fly from A to B at 200 km/h, then return from B to A at the same speed, would the total trip be longer, the same, or shorter than if you had a headwind of 90 knots? The answer is: the round trip is always shorter when there is no wind. The same applies for crosswind as in your example. because if you move to the NNE, you destination is still to the N and you crab sideways, thus over a longer period of time.

PS: Pilots use a special slide rule to graphically solve these wind component problems.