Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 84 / 6 / d / Solution

If $h>0$, the front enters the lower-opening branch before reaching the floor. Near $z_f=0$,The final approach is exponential and the time integral in part (c) diverges logarithmically. Thus no finite floor-arrival time exists. If the opening is at $h=0$, the first branch applies until arrival and has a finite nonzero downward speed at the floor. The arrival time isHence finite-time floor arrival occurs only for a floor opening, within the stated finite-source model. In the point-source limit $Q_s\to 0$, $z_v\to 0$ and this arrival time diverges, recovering part (b).