Question

An aeroplane flies along a straight path A to B and returns back again. The distance between A and B is $l$ and the aeroplane maintains the constant speed v w.r.t. wind. There is a steady wind with a speed u at an angle $\theta$ with line AB. Determine the expression for the total time of the trip.

Answer 1

Suppose the plane is orented at an angle $\alpha$w.r.t line AB.while

the plane is moving from $A$ to $B$

Velocity of plane along

$AB=v\cos \alpha -u\cos \theta$ and for no - drift from line $AB$

$v\sin \alpha =u\sin \theta$

$\Rightarrow \sin \alpha =\frac{u\sin \theta }{v}$

Time taken from $A$ to $B$ :$t_{AB}=\frac{t}{v\cos \alpha -u\cos \theta }$

Suppose plane is orented at an angle $alpha$ w.r.t line $AB$ while the plane is moving from $B$ to $A$

velocity of plane along $BA$=$v\cos \alpha +u\cos \theta$ and for no drift from line $AB$.

$v\sin \alpha =u\sin \theta$

$\Rightarrow \sin \alpha =\frac{u\sin \theta }{v}\to \alpha ={\alpha }^{\prime }$

Time taken from $B$ to $A$;

$t_{BA}=\frac{l}{v\cos \alpha +u\cos \theta }$

Total time taken $=tAB+t_{BA}$

$=\frac{l}{v\cos \alpha }+\frac{l}{v\cos \alpha +u\cos \theta }$

$\frac{2vl\cos \alpha }{v^2{\cos }^2\alpha +u^2{\cos }^2\theta }$

$=\frac{2vt\sqrt{1-\frac{u^2{\sin }^2\theta }{v^2}}}{v^2-u^2}$