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(A) To travel from $A$ to $B$ in a straight line while a wind of speed $v$ blows perpendicular to $AB$,the aeroplane must head at an angle such that i

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$\frac{2\ell}{\sqrt{V^2 - v^2}}$

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(A) To travel from $A$ to $B$ in a straight line while a wind of speed $v$ blows perpendicular to $AB$,the aeroplane must head at an angle such that its resultant velocity vector points directly along $AB$.Let the resultant velocity of the aeroplane relative to the ground be $V_g$.Using the Pythagorean theorem,$V^2 = V_g^2 + v^2$,which gives the ground speed $V_g = \sqrt{V^2 - v^2}$.The time taken to travel from $A$ to $B$ is $t_1 = \frac{\ell}{V_g} = \frac{\ell}{\sqrt{V^2 - v^2}}$.Similarly,the time taken to return from $B$ to $A$ is $t_2 = \frac{\ell}{V_g} = \frac{\ell}{\sqrt{V^2 - v^2}}$.The total time for the round trip is $T = t_1 + t_2 = \frac{\ell}{\sqrt{V^2 - v^2}} + \frac{\ell}{\sqrt{V^2 - v^2}} = \frac{2\ell}{\sqrt{V^2 - v^2}}$.

An aeroplane is to go along a straight line from $A$ to $B$,and back again. The relative speed with respect to wind is $V$. The wind blows perpendicular to the line $AB$ with speed $v$. The distance between $A$ and $B$ is $\ell$. The total time for the round trip is:

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