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.

\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}}

Kinematics