Physics help: Traffic shock wave?

Question

An abrupt slowdown in concentrated traffic can travel as a pulse, termed a shock wave, along the line of cars, either downstream (in the traffic direction) or upstream, or it can be stationary. figure below shows a uniformly spaced line of cars moving at speed v = 25.0 m/s toward a uniformly spaced line of slow cars moving at speed v_s = 5.00 m/s. Assume that each faster car adds length L = 12.0 m (car length plus buffer zone) to the line of slow cars when it joins the line, and assume it slows abruptly at the last instant. (a) For what separation distance d between the faster cars does the shock wave remain stationary? If the separation is twice that amount, what are the (b) speed and (c) direction (upstream or downstream) of the shock wave?

Show work, please.

Answer 1

Since the slow zone is moving at 5m/s, if the shock front is to remain stationary, then there must be 5m/s of cars being added to it. Each car will add 12m to the length of the slow zone, so a car must join the slow zone once every 2.4 seconds (12/5). Given that time separation, and the speed of 25m/s for the fast zone, the cars must be separated by 60m (25*2.4).

If the cars are separated by twice this distance, then 12m get added to the end of the shock zone every 4.8s (120/25). So that comes out to 2.5m/s (12/4.8). Thus the shockwave advances down the roadway at 2.5m/s (5-2.5),