Problem 1.25

A point moves in the plane $x y$ according to the law $x=a t$ , $y=$ at $(1 \stackrel{-\alpha t}{t}),$ where $a$ and $\alpha$ are positive constants, and $t$ is time. Find: (a) the equation of the point's trajectory $y(x)$ ; plot this function; (b) the velocity $v$ and the acceleration $w$ of the point as functions of time; (c) the moment $t_{0}$ at which the velocity vector forms an angle $\pi / 4$ with the acceleration vector.

\text { (a) } y=x-x^{2} \alpha / a \text { (b) } v=a \sqrt{1+(1-2 \alpha t)^{2}}, w=2 \alpha a= \begin{aligned} &=\text { const; }\ &\text { (c) } t_{0}=1 / \alpha \end{aligned}

Kinematics