Solve the problem.
-Given the velocity and initial position of a body moving along a coordinate line at time $t$, find the body's position at time $t$.
$$v = \frac { 8 } { \pi } \sin \frac { 4 t } { \pi } , s \left( \pi ^ { 2 } \right) = 2$$
A) $s = - 2 \cos \frac { 4 t } { \pi } + 3$
B) $\mathrm { s } = - 2 \cos \frac { 4 \mathrm { t } } { \pi } + 4$
C) $s = 2 \cos \frac { 4 t } { \pi } + 4$
D) $\mathrm { s } = - 2 \cos \frac { 4 \mathrm { t } } { \pi } + 8$

Question

Quizplus · Accepted Answer

To find the position $s(t)$ given the velocity $v(t) = \frac{8}{\pi} \sin\left(\frac{4t}{\pi}\right)$, we integrate $v(t)$ with respect to $t$. The integral of $\sin\left(\frac{4t}{\pi}\right)$ is $-\frac{\pi}{4}\cos\left(\frac{4t}{\pi}\right) + C$, where $C$ is the constant of integration. Multiplying by $\frac{8}{\pi}$, we get $s(t) = -2\cos\left(\frac{4t}{\pi}\right) + C$. To find $C$, we use the initial condition $s(\pi^2) = 2$, which gives $2 = -2\cos(4\pi) + C$. Since $\cos(4\pi) = 1$, we have $2 = -2(1) + C$, so $C = 4$. Therefore, the position function is $s(t) = -2\cos\left(\frac{4t}{\pi}\right) + 4$.

Solve the Problem ttt , Find the Body's Position at Time ttt

Solve the Problem $t$ , Find the Body's Position at Time $t$