David wants to fly his plane due north at 310 km/h relative to the ground. If a 80 km/h wind is blowing from the north-west, what velocity in the air should he fly his plane?
Give your answer in terms of the unit vectors i and j, where north corresponds to the positive y direction.
east wind particle velocity = 80 cos 45
north wind particle velocity = - 80 cos 45
S = Plane air speed
A =angle plane aims west of north
west plane velocity = east wind particle velocity = 80 cos45 = 56.6 km/h
if S =airspeed
- S sin A = -56.6
sin A = 56.6/S
sin^2 A = 3203/S^2
northward now
S cos A - 56.6 = 310 so S cos A = 366.6
cos A = 366.6 / S
cos^2 A = 134395/S^2
but we know sin^2 + cos^2 = 1
so
3203/S^2 + 134395/S^2 = 1
S^2 = 137598
S = 371 km/hr
sin A = 56.6/S
A = 8.78 degrees
so
V = -371 sin A i + 371 cos A j = - 56.6 i + 366.6 j