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A balloon starts rising from earth surface with constant speed v₀. Due to wind the balloon gathers a horizontal velocity component v_x= ay where a is a positive constant and y is its height above earth surface. Find the horizontal drift of balloon as a function of height.

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$\frac{a}{v_{0}} y^{2}$

$\frac{a}{2 v_{0}} y^{2}$

$\frac{a}{v_{0}} y^{3}$

$\frac{a}{3 v_{0}} y^{2}$

answer is B.

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Detailed Solution

due to uniform speed in y direction,

its height as a function of time= y=v₀t

In x-distance $v_{x} = ay = {av}_{0} t \Rightarrow t = \frac{y}{v_{0}}$

$\frac{dx}{dt} = {av}_{0} t$

$\int_{0}^{x} dx = \int_{0}^{t} {av}_{0} tdt$

$x = \frac{{av}_{0} t^{2}}{2}$

$x = \frac{{av}_{0}}{2} {(\frac{y}{v_{0}})}^{2} = \frac{a}{2 v_{0}} y^{2}$

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