Answer
a) v(t)=sin(t), s(t)=-cos(t)+4
b) t=k$\pi$ where k=0,1,2,3... ($k∈\mathbb{Z}^{+}$)
Work Step by Step
a) Let a(t) = cos(t)
v=$\int$a dt
v=$\int$cos(t) dt
v=sin(t) +$c_{1}$ where $c_{1}$ is an arbitrary constant
When t=0, v=0 since the particle is initially at rest
∴v(t)=sin(t)
s=$\int$v dt
s=$\int$sin(t) dt
s=-cos(t) +$c_{2}$ where $c_{2}$ is an arbitrary constant
When t=0, s=3
3=cos(0)+$c_{2}$
3=-1+$c_{2}$
$c_{2}$=4
∴s(t)=-cos(t)+4
b) For particle to be at rest, v(t)=0
∴sin(t)=0
t=sin$^{-1}$(0)
t=$\pi, 2\pi, 3\pi, ....$
t=k$\pi$ where k=0,1,2,3...
