Razzle Dazzle

A particle in the \(xy\)-plane has the following velocity and experiences the following force: \[\begin{align} \vec{v} &= v_x \hat{\imath} + v_y \hat{\jmath} \hspace{1cm} (\text{m/s})\\ \\ \vec{F} &= e^{-t} \sin\left(t + \frac{\pi}{2}\right) \left[e^{i\pi} v_y \hat{\imath} + e^{i0} v_x \hat{\jmath}\right]. \hspace{1cm} (\text{N}) \end{align}\] Suppose the particle has an initial velocity of \((v_x,v_y) = (3\text{ m/s}, 4 \text{ m/s})\) at time \(t=0\text{ s}\).

What is the total path length covered by the particle between \(t = 0 \text{ s}\) and \(t = 1 \text{ s}?\)

\(\)
Note: The symbol \(i\) denotes the imaginary unit. Symbols \(\hat{\imath}\) and \(\hat{\jmath}\) represent unit vectors in the \(x\)- and \(y\)-directions, respectively.

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