# 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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