The distance ss, in meters, of an object from the origin at time t≥0t≥0 seconds is given by s=s(t)=Acos(ωt+ϕs=s(t)=Acos(ωt+ϕ), where A,A, ω,ω, and ϕϕ are constant.
(a) Find the velocity vv of the object at time tt.
(b) When is the velocity of the object 00?
(c) Find the acceleration aa of the object at time t.t.
(d) When is the acceleration of the object 0?
+26387 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.
\(\small{ \begin{array}{lrcll} & s(t) &=& A\cdot cos(\omega t+\phi) \quad & ( \text{distance} ) \\ (a) \text{ Find the velocity v of the object at time t. } \\ & v(t) &=& \frac{ds}{dt} = -A\cdot \omega \cdot sin(\omega t+\phi) \quad & ( \text{velocity} ) \\ (c) \text{ Find the acceleration a of the object at time t.}\\ & a(t) &=& \frac{d^2s}{dt^2} = -A\cdot \omega^2 \cdot cos(\omega t+\phi)\quad & ( \text{acceleration} ) \end{array} }\)
\((b) \text{ When is the velocity of the object 0?}\\ \begin{array}{rcll} v(t) = -A\cdot \omega \cdot sin(\omega t+\phi) &=& 0 \\ sin(\omega t+\phi) &=& 0 \\ \omega t+\phi &=& \arcsin{(0)} \pm k\cdot \pi \\ \omega t+\phi &=& 0 \pm k\cdot \pi \\ \omega t &=& -\phi \pm k\cdot \pi \\ t &=& \frac{ -\phi \pm k\cdot \pi }{ \omega } \qquad k = 0,1,2,\dots\\ \end{array}\)
\((d) \text{ When is the acceleration of the object 0?}\\ \begin{array}{rcll} a(t) = -A\cdot \omega^2 \cdot cos(\omega t+\phi) &=& 0\\ cos(\omega t+\phi) &=& 0 \\ \omega t+\phi &=& \arccos{(0)} \pm k\cdot \pi \\ \omega t+\phi &=& \frac{\pi}{2} \pm k\cdot \pi \\ \omega t &=& \frac{\pi}{2} -\phi \pm k\cdot \pi \\ t &=& \frac{ \frac{\pi}{2} -\phi \pm k\cdot \pi }{ \omega } \qquad k = 0,1,2,\dots\\ \end{array}\)
+26387 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.
\(\small{ \begin{array}{lrcll} & s(t) &=& A\cdot cos(\omega t+\phi) \quad & ( \text{distance} ) \\ (a) \text{ Find the velocity v of the object at time t. } \\ & v(t) &=& \frac{ds}{dt} = -A\cdot \omega \cdot sin(\omega t+\phi) \quad & ( \text{velocity} ) \\ (c) \text{ Find the acceleration a of the object at time t.}\\ & a(t) &=& \frac{d^2s}{dt^2} = -A\cdot \omega^2 \cdot cos(\omega t+\phi)\quad & ( \text{acceleration} ) \end{array} }\)
\((b) \text{ When is the velocity of the object 0?}\\ \begin{array}{rcll} v(t) = -A\cdot \omega \cdot sin(\omega t+\phi) &=& 0 \\ sin(\omega t+\phi) &=& 0 \\ \omega t+\phi &=& \arcsin{(0)} \pm k\cdot \pi \\ \omega t+\phi &=& 0 \pm k\cdot \pi \\ \omega t &=& -\phi \pm k\cdot \pi \\ t &=& \frac{ -\phi \pm k\cdot \pi }{ \omega } \qquad k = 0,1,2,\dots\\ \end{array}\)
\((d) \text{ When is the acceleration of the object 0?}\\ \begin{array}{rcll} a(t) = -A\cdot \omega^2 \cdot cos(\omega t+\phi) &=& 0\\ cos(\omega t+\phi) &=& 0 \\ \omega t+\phi &=& \arccos{(0)} \pm k\cdot \pi \\ \omega t+\phi &=& \frac{\pi}{2} \pm k\cdot \pi \\ \omega t &=& \frac{\pi}{2} -\phi \pm k\cdot \pi \\ t &=& \frac{ \frac{\pi}{2} -\phi \pm k\cdot \pi }{ \omega } \qquad k = 0,1,2,\dots\\ \end{array}\)
