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Kinematics

Problem 1.34

A balloon starts rising from the surface of the Earth. The ascension rate is constant and equal to \(v_{0}\). Due to the wind the balloon gathers the horizontal velocity component \(v_{x}=a y,\) where \(a\) is a constant and \(y\) is the height of ascent. Find how the following quantities depend on the height of ascent: (a) the horizontal drift of the balloon \(x(y)\); (b) the total, tangential, and normal accelerations of the balloon.

Reveal Answer
 (a) x=(a/2v0)y2 ;  (b) w=av0,wτ=a2y/1+(ay/v0)2wn=av0/1+(ay/v0)2\text { (a) } x=\left(a / 2 v_{0}\right) y^{2} \text { ; } \text { (b) } w=a v_{0}, \quad w_{\tau}=a^{2} y / \sqrt{1+\left(a y / v_{0}\right)^{2}} w_{n}=a v_{0} / \sqrt{1+\left(a y / v_{0}\right)^{2}}
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