In a laboratory experiment, a sphere of diameter \(8.0 \mathrm{~mm}\) is released from rest at \(t=0\) at the surface of honey in a jar, and the sphere's downward speed \(v\) when it travels in the honey is found to be given by \(v=v_{\max }\left(1-e^{-t / \tau}\right)\), where \(v_{\max }=0.040 \mathrm{~m} / \mathrm{s}\) and \(\tau=0.50\) s.

(a) Obtain an expression for \(a(t)\).

(b) Draw graphs for \(v(t)\) and \(a(t)\) for the time interval 0 to \(2.0 \mathrm{~s}\).

(c) Obtain an expression for \(x(t)\), choosing the positive \(x\) axis as downward, and draw the graph for this function.

(d) Use your \(x(t)\) graph to determine the time interval needed for the sphere to reach the bottom of the container if the surface of the honey is \(0.10 \mathrm{~m}\) above the bottom of the jar.