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

Question

Quizplus · Accepted Answer

We are given the velocity function $v = \cos \frac{\pi}{2} t$ and the initial position $s(0) = 1$. We can start by finding the position function by integrating the velocity function:

$$s(t) = \int v(t) dt = \int \cos \frac{\pi}{2} t dt = \frac{2}{\pi} \sin \frac{\pi}{2} t + C$$

To find the value of the constant $C$, we use the initial condition $s(0) = 1$:

$$1 = \frac{2}{\pi} \sin 0 + C \quad \Rightarrow \quad C = 1$$

Substituting this value back into the position function, we get:

$$s(t) = \frac{2}{\pi} \sin \frac{\pi}{2} t + 1$$

Simplifying this expression, we find that it matches choice C: $s(t) = \frac{2}{\pi} \sin \frac{\pi}{2} t + 1$. Therefore, the answer is choice 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 $\mathrm { t }$ $v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1$