The \(x\) - and \(y\)-motions of guides \(A\) and \(B\) with rightangle 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 \(t=2 \mathrm{~s}\). Sketch the direction of the path and indicate its curvature for this instant.

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P B A y X