Meriam, Kraige & Bolton — Kinematics of Particles: Problem 2_50

2/50 The $x$- and $y$-motions of guides $A$ and $B$ with right-angle slots control the curvilinear motion of the connecting pin $P$, which slides in both slots. For a short interval, the motions are governed by $x = 20 + \frac{1}{4}t^2$ and $y = 15 - \frac{1}{6}t^3$, where $x$ and $y$ are in millimeters and $t$ is in seconds. Calculate the magnitudes of the velocity $\mathbf{v}$ and acceleration $\mathbf{a}$ of the pin for $t = 2\text{ s}$. Sketch the direction of the path and indicate its curvature for this instant.