A video recording of a moving object shows the object moving up and down with a velocity given by \(v_{x}(t)=\) \(v_{\max } \cos (\omega t)\), where \(v_{\max }=1.20 \mathrm{~m} / \mathrm{s}\) and \(\omega=0.15 \mathrm{~s}^{-1}\) and where the positive \(x\) direction is upward.

(a) Derive an expression for \(a(t)\), the object's acceleration as a function of time.

(b) If the object's position at \(t=0\) is \(x(0)=0\), derive an expression for \(x(t)\), its position as a function of time.