A horizontal conveyor belt moves at a constant speed \(v_0\). Sand is dropped vertically from rest onto the belt at a time-dependent rate given by \(\dfrac{dm}{dt} = kt\), where \(k\) is a positive constant and \(t\) is time. Friction between the sand and the belt quickly accelerates the sand horizontally to match the speed of the belt. Which of the following expressions represents the total work done by the motor driving the belt during the time interval from \(t = 0\) to \(t = T\)?