Question

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?

• A. $\theta =ta{n}^{-1}\left(\frac{1}{2\sqrt{2}}\right)$
  $v_{\frac{p}{g}}=200\sqrt{2}m/s$
  
• B. $\theta =ta{n}^{-1}\left(1\right)$ $v_{\frac{p}{g}}=100m/s$.
• C. $\theta =ta{n}^{-1}\left(\frac{1}{3}\right)$ $v_{\frac{p}{g}}=300m/s$.
• D. $\theta =ta{n}^{-1}\left(\frac{2}{3}\right)$ $v_{\frac{p}{g}}=200\sqrt{2}m/s$.

Answer 1

The correct option is A $\theta =ta{n}^{-1}\left(\frac{1}{2\sqrt{2}}\right)$

$v_{\frac{p}{g}}=200\sqrt{2}m/s$

GIven velocity of air with respect to ground $v_{\frac{A}{G}}=100km/hr=-100\hat{j}$

Velocity of plane with respect to air ${\overrightarrow{v}}_{\frac{P}{A}}=300km/hr=-300cos\theta \hat{i}+300sin\theta \hat{j}$

(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 $\theta$ towards north-west direction, in order to reach at point on west.)

${\overrightarrow{v}}_{\frac{P}{A}}={\overrightarrow{v}}_{\frac{P}{G}}-{\overrightarrow{v}}_{\frac{A}{G}}\Rightarrow {\overrightarrow{v}}_{\frac{P}{G}}={\overrightarrow{v}}_{\frac{P}{G}}+{\overrightarrow{v}}_{\frac{A}{G}}=-300cos\theta \hat{i}+300sin\theta \hat{j}+-100\hat{j}$

$v_{\frac{P}{G}}=-300cos\theta \hat{i}+(300sin\theta -100)\hat{j}$

We know that the plane has to go west wards and so the component of velocity towards north or in $\hat{j}$

should be 0

$\Rightarrow 300sin\theta -100=0$

$sin\theta =\frac{1}{3}$

$\Rightarrow cos\theta =\frac{2\sqrt{2}}{3}$

$tan\theta =\frac{1}{2\sqrt{2}}$

$\theta =ta{n}^{-1}\left(\frac{1}{2\sqrt{2}}\right)$

Direction heading of plane

$\Rightarrow v_{\frac{P}{G}}=-300cos\theta \hat{i}(\hat{j}=0)$

$\Rightarrow v_{\frac{P}{G}}=-300\times \frac{2\sqrt{2}}{3}$

$v_{\frac{P}{G}}=-200\sqrt{2}\hat{i}$

So the plane will appear to be heading west with a velocity of $200\sqrt{2}$ m/s.