Parametric Differentiation: Velocity and Acceleration Analysis
Consider the parametric equations for the motion of a particle given by \(x(t) = t^2\) and \(y(t) = e^{-t}\). (a) Determine the velocity vector \(\mathbf{v}(t)\) of the particle as a function of \(t\). (b) Differentiate further to find the acceleration vector \(\mathbf{a}(t)\) in parametric form.