An airplane pilot wants to deliver a package to a town 60 miles east and 100 mil
ID: 1430603 • Letter: A
Question
An airplane pilot wants to deliver a package to a town 60 miles east and 100 miles north of his current location. The wind is blowing to the east at 30mi/hr. If the plane is capable of flying at an airspeed of 100mi/hr (i.e. a speed of 100mi/hr relative to the air), what should be the heading of the plane? (The heading is the direction the plane points in, and is the direction of the plane’s velocity vector relative to the air.) Your answer should be stated in terms of compass directions.
How do i come up with the 3 vectors and how do i label vectors like whats the reasoning behind the velocity of the plane w respect to the ground.
Explanation / Answer
airplane wants to go to town hence (i is along east and j is along north)
displacement vector, r = 60i + 100j mile
velocity of wind relative to earth = 30i mph
speed of plane relative to air = 100 mph
plane want to go to town. so we observed plane relative ro earth.
so velocity of plane wrt earth must be along the displacement vector, r = 60i+ 100j
velocity of plane relative to air = velocity of plane - velocity of air
velocity of plane = velocity of plane relative to air + velocity of air
and suppose plane is headed @ angle with east in north east.
then Vp = 100 (cos@ i + sin@ j) + 30i
Vp = (100cos@ + 30)i + 100sin@ j
direction of Vp and 60i +100j are same.
angle made by 60i +100j with east
angle = tan^-1(100/60) = 59.04 deg
same angle VP made with east
so tan59.04 = (100 sin@ )/ (100cos@ + 30)
500cos@ + 150 = 300sin@
square both side
2500 (cos@)^2 + 225 = 900 (sin@)^2 = 900 (1 - cos^2 @)
1600 cos^2@ + 1500cos@ - 675 = 0
cos@ = 0.332
@ = 70.60 deg north to east ( angle with east direction in northeast)