(Solved): Let \( r(t) \) and \( \theta(t) \) be polar coordinates which are functions of a parameter \( t \) ...

Let \( r(t) \) and \( \theta(t) \) be polar coordinates which are functions of a parameter \( t \). (a) Express \( \frac{d x}{d t} \) and \( \frac{d y}{d t} \) in terms of \( \frac{d r}{d t}, \frac{d \theta}{d t}, r \), and \( \theta \). (b) Prove that \[ \begin{array}{l} \cos \theta \frac{d^{2} x}{d t^{2}}+\sin \theta \frac{d^{2} y}{d t^{2}}=\frac{d^{2} r}{d t^{2}}-r\left(\frac{d \theta}{d t}\right)^{2} \\ \cos \theta \frac{d^{2} y}{d t^{2}}-\sin \theta \frac{d^{2} x}{d t^{2}}=\frac{1}{r} \frac{d}{d t}\left(r^{2} \frac{d \theta}{d t}\right)^{2} \end{array} \] (These two equations express the radial and tangential co of acceleration, given on the left, in terms of polar coordin