Question

An aircraft starts from point $A$ and it has to reach point $B$. Wind is blowing with speed $y$ as shown in the figure. The aircraft flies with speed $v$ with respect to the wind. The aircraft starts from $A$ making an angle $\alpha$ with $AB$. Find the correct relation between $v$ and $y$ for the aircraft to reach point $B$.

• A. $v\cos 2\theta =y\cos 2\alpha$
• B. $v\sin \theta =y\sin \alpha$
• C. $v\cos \theta =y\cos \alpha$
• D. $v\cos \alpha =y\cos \theta$

Answer 1

The correct option is D $v\cos \alpha =y\cos \theta$

We know
${\overrightarrow{v}}_{Ag}={\overrightarrow{v}}_{A\omega }+{\overrightarrow{v}}_{\omega g}$
where all the terms have standard meanings.

Now, putting values according to question :
${\overrightarrow{v}}_{Ag}=(v\cos \alpha \hat{i}+v\sin \alpha \hat{j})+(-y\cos \theta \hat{i}-y\sin \theta \hat{j})$

For the resultant velocity of aircraft to be oriented along $AB$, the $y$ component must be $0$.
i.e $v\sin \alpha -y\sin \theta =0$
$\Rightarrow v\sin \alpha =y\sin \theta$