Halliday, Resnick & Walker — Motion Along a Straight Line: Problem 12

Traffic shock wave. 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 2.10 shows a uniformly spaced line of cars moving at speed \(v = 25.0 \text{ m/s}\) toward a uniformly spaced line of slow cars moving at speed \(v_s = 5.00 \text{ m/s}\). Assume that each faster car adds length \(L = 12.0 \text{ m}\) (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?