Question

an object’s velocity at time t is given by v(t) = –2 sin t. Let s(t) represent the object’s position at time t. If s(0) = 0, then s(t) =

Answer 1

GIVEN

The function of the object's velocity is given as follows:

`v(t)=-2\sin t`

Also given:

`s(0)=0`

SOLUTION

To get the position's function (s(t)), the velocity function needs to be integrated:

`s(t)=\int v(t)dt`

Therefore:

`\begin{gathered} s(t)=\int(-2\sin t)dt \\ \mathrm{Take\:the\:constant\:out}: \\ s(t)=-2\cdot\int\sin\left(t\right)dt \\ \mathrm{Use\:the\:common\:integral}:\quad \int \sin \left(t\right)dt=-\cos \left(t\right) \\ s(t)=-2\left(-\cos\left(t\right)\right) \\ \mathrm{Simplify}\text{ and add a constant to the solution} \\ s(t)=2\cos\left(t\right)+C \end{gathered}`

Recall that s(0) = 0. Therefore:

`\begin{gathered} s(0)=2\cos(0)+C=0 \\ \therefore \\ C=-2 \end{gathered}`

Hence, the position function is:

`s(t)=2\cos t-2`

The THIRD OPTION is correct.