Question

An aeroplane pilot wishes to fly due west. A wind of $100km{h}^{-1}$ is blowing towards south.
a. If the speed of the plane (its speed in still air) is $300km{h}^{-1}$, in which direction should the pilot head?
b. What is the speed of the plane with respect to ground? Illustrate with a vector diagram.

Answer 1

Velocity of air (wind) = $v_A=100km{h}^{-1}$
Velocity of plane w.r.t. air = $V_{P/A}=300km{h}^{-1}$
${\overrightarrow{v}}_P={\overrightarrow{v}}_{P/A}+{\overrightarrow{v}}_A$
The net velocity of the plane will be the vector sum of two velocities.
Velocity of air and velocity of plane w.r.t air. If the plane is to move towards west finally, then the N-S component of velocity should be zero. For this,
${\overrightarrow{v}}_{P/A}sin\theta =v_A$
$\Rightarrow 300\sin \theta =100$
$\Rightarrow \sin \theta =\frac{1}{3}\Rightarrow \theta ={\sin }^{-1}\left(\frac{1}{3}\right)$
So the pilot should head in direction $\theta ={\sin }^{-1}\left(\frac{1}{3}\right)$ N of W.
Speed of plane w.r.t ground,
$v_P=v_{P/A}\cos \theta$
$=300\sqrt{1-si{n}^2\theta }=300\sqrt{1-{\left(\frac{1}{3}\right)}^2}=200\sqrt{2}km{h}^{-1}$