Solve the problem.
-Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's positior $\mathrm { t }$.
$v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1$
A) $s = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } t + \pi$
B) $s = 2 \pi \sin \frac { \pi } { 2 } t$
C) $s = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } t + 1$
D) $s = \sin t + 1$ time

Question

Quizplus · Accepted Answer

Since the velocity is given as $v=\cos\frac{\pi}{2}t$, we can use integration to find the position of the body at time t. Integrating v with respect to t gives us the position function $s(t)$.
$$  s(t) = \int v dt = \int \cos \frac{\pi}{2} t dt = \frac{2}{\pi} \sin\frac{\pi}{2}t + C$$
where $C$ is an arbitrary constant. To find the value of $C$, we use the initial position $s(0)=1$. 
$$ s(0) = \frac{2}{\pi} \sin\frac{\pi}{2}(0) + C = 1$$
Hence $C=1$. 
Therefore, the position of the body at time t is given by:
$$ s(t) = \frac{2}{\pi} \sin\frac{\pi}{2}t + 1$$
which matches with option C.

Solve the Problem t\mathrm { t }t v=cos⁡π2t,s(0)=1v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1v=cos2π​t,s(0)=1

Solve the Problem $\mathrm { t }$ $v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1$