An aero-plane pilot wishes to fly due west. A wind of 100 km/h is blowing toward the south. If the airspeed of the plane (its speed in still air) is 300 km/h, I which direction should the pilot head? What is the speed of the plane w.r.t. the ground?
GIven velocity of air with respect to ground vAG=100km/hr=−100^j
Velocity of plane with respect to air →vPA=300km/hr=−300cosθ^i+300sinθ^j
(As the plane is to move towards west, due to air in south direction,air drift the plane in south direction.
Hence plane has to make an angle θ towards north-west direction, in order to reach at point on west.)
→vPA=→vPG−→vAG⇒→vPG=→vPG+→vAG=−300 cosθ^i+300 sinθ^j+−100^j
vPG=−300 cos θ^i+(300 sinθ−100)^j
We know that the plane has to go west wards and so the component of velocity towards north or in ^j
should be 0
⇒300 sinθ−100=0
sinθ=13
⇒cosθ=2√23
tanθ=12√2
θ=tan−1(12√2)
Direction heading of plane
⇒vPG=−300 cosθ^i(^j=0)
⇒vPG=−300×2√23
vPG=−200√2^i
So the plane will appear to be heading west with a velocity of 200√2 m/s.