# Vector (Kinematic)

(CIE 0606/2021/m/22/Q12)
A particle $P$ travels in a straight line so that, $t$ seconds after passing through a fixed point $O$, its velocity, $v \mathrm{~ms}^{-1}$, is given by

$v=\dfrac{t}{2 \mathrm{e}}$ for $0 \leqslant t \leqslant 2$,

$v=\mathrm{e}^{-\dfrac{t}{2}} \quad$ for $t>2$.

Given that, after leaving $O$, particle $P$ is never at rest, find the distance it travels between $t=1$ and $t=3 .$

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$\begin{array}{ll} &\text { The distance it travels between } t=1 \text { and } t=3 \\&=\displaystyle\int_{1}^{2} \dfrac{t}{2 e} d t+\displaystyle\int_{2}^{3} e^{-\frac{t}{2}} d t \\&=\dfrac{1}{2e} \left[\dfrac{t^{2}}{2} \right]_{1}^{2}+\left[ \dfrac{e^{- \frac{t}{2}}}{\left(-\frac{1}{2}\right)}\right]_{2}^{3} \\&=\dfrac{1}{4 e}\left(2^{2}-1^{2}\right)-2\left(e^{-\frac{3}{2}}-e^{-1}\right) \\&=0.565\end{array}$
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