The distance s, in meters, of an object from the origin at time t=0
seconds is given by\( s=s(t)=A\cdot cos(\omega t+\phi)\), where A, \(\omega\), and \(\phi\) are constant.
\(\small{ \begin{array}{lrcll} & s(t) &=& A\cdot cos(\omega t+\phi) \quad & ( \text{distance} ) \\ (a) \text{ Find the velocity v of the object at time t. } \\ & v(t) &=& \frac{ds}{dt} = -A\cdot \omega \cdot sin(\omega t+\phi) \quad & ( \text{velocity} ) \\ (c) \text{ Find the acceleration a of the object at time t.}\\ & a(t) &=& \frac{d^2s}{dt^2} = -A\cdot \omega^2 \cdot cos(\omega t+\phi)\quad & ( \text{acceleration} ) \end{array} }\)
\((b) \text{ When is the velocity of the object 0?}\\ \begin{array}{rcll} v(t) = -A\cdot \omega \cdot sin(\omega t+\phi) &=& 0 \\ sin(\omega t+\phi) &=& 0 \\ \omega t+\phi &=& \arcsin{(0)} \pm k\cdot \pi \\ \omega t+\phi &=& 0 \pm k\cdot \pi \\ \omega t &=& -\phi \pm k\cdot \pi \\ t &=& \frac{ -\phi \pm k\cdot \pi }{ \omega } \qquad k = 0,1,2,\dots\\ \end{array}\)
\((d) \text{ When is the acceleration of the object 0?}\\ \begin{array}{rcll} a(t) = -A\cdot \omega^2 \cdot cos(\omega t+\phi) &=& 0\\ cos(\omega t+\phi) &=& 0 \\ \omega t+\phi &=& \arccos{(0)} \pm k\cdot \pi \\ \omega t+\phi &=& \frac{\pi}{2} \pm k\cdot \pi \\ \omega t &=& \frac{\pi}{2} -\phi \pm k\cdot \pi \\ t &=& \frac{ \frac{\pi}{2} -\phi \pm k\cdot \pi }{ \omega } \qquad k = 0,1,2,\dots\\ \end{array}\)
