Let \(\mathcal{G}\) be the graph of the parametric equations \(\begin{align*} x&=1+\sin s,\\ y&=2-\cos(2s) \end{align*}\) for all \(s\) in the interval \([0,\pi)\).
To what interval must we restrict the parameter \(t\) if the graph of the parametric equations \(\begin{align*} x&=t+1,\\ y&=2t^2 +1 \end{align*}\)
for \(t\) in that interval produces the same graph \(\mathcal{G}\)?