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 $t$.
$$\mathrm { v } = \cos \frac { \pi } { 2 } \mathrm { t } , \mathrm { s } ( 0 ) = 1$$
A) $s = \sin t$
B) $s = 2 \pi \sin \frac { \pi } { 2 } t$
C) $s = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } t + \pi$
D) $\mathrm { s } = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } \mathrm { t }$

Question

Quizplus · Accepted Answer

The position $s(t)$ can be found by integrating the velocity function $v(t) = \cos \frac{\pi}{2}t$. Integrating gives $s(t) = \frac{2}{\pi}\sin\frac{\pi}{2}t + C$. Using the initial condition $s(0) = 1$, we find $C = 1$, so the correct expression is $s(t) = \frac{2}{\pi}\sin\frac{\pi}{2}t + 1$. However, none of the options exactly match this, but option D is the closest to the correct form, missing only the constant term, which was likely an oversight in the question or answer choices.

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