ANSWER
4 ft
EXPLANATION
The velocity equation of the oscillating spring is given by the function.
[tex]v(t)=2 \sin(t) \: \: {ms}^{ - 1} [/tex]
To find the displacement function, we need to to Integrate the velocity function.
[tex] s(t) = \int \: 2 \sin(t) dt[/tex]
[tex]s(t) = - 2 \cos(t) + k[/tex]
At time t=0, there was no displacement.
This implies that,
[tex]s(0) = 0[/tex]
[tex]0= - 2 \cos(0) + k[/tex]
[tex]0= - 2 + k[/tex]
[tex]k = 2[/tex]
The displacement function then becomes,
[tex]s(t) = - 2 \cos(t) + 2[/tex]
To find the displacement over the first π seconds, we put
[tex]t = \pi[/tex]
into the equation for the displacement to get,
[tex]s(\pi) = - 2 \cos(\pi) + 2[/tex]
[tex]s(\pi) = - 2 ( - 1) + 2[/tex]
[tex]s(\pi) = 2 + 2 =4 ft[/tex]
