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. The figure 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}=25.0 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?

#### Solution:

### a) Let d represent the distance, between the fast and slow cars, at t=0. The time it takes for a slow car to move 12.0m is t=\frac{L}{v_{s}}=\frac{12.0 m}{5.0 m/s}= 2.40 s. During the 2.4 seconds, the fast car movies a distance of vt=d+L and joins the line of slow cars, which means the shock wave remains stationary. This also means there is a separation of:

### d=vt-L =(25m/s)(2.4 s) - 12.0 m=48.0 m .

### b) If the separation is twice that amount, it is equal to 48 m \times 2= 96.0 m at t=0 . As time passes, the total distance the slow and fast cars travel can be represented as x=v_{s}t and the fast cars join the line by moving a distance of d+x . This means:

### t=\frac{x}{v_{s}}= \frac{d+x}{v}.

### From this, we can derive that:

### x= \frac{v_{s}}{v-v{s}}d= \frac{5.00m/s}{25.0m/s - 5.00 m/s}(96.0 m) = 24 m.

### Therefore, t= \frac{(24.0 m)}{ (5.00 m/s)} = 4.80 s . To find the speed of the shock wave, it is important to understand that the distance the slow cars have moved is equal to \triangle x=x-L= 24.0 m- 12.0m = 12.0 m . Therefore, speed of the shock wave is:

### v=\frac{\triangle x}{t}= \frac{12.0 m}{4.80 s}=2.50 m/s

### c) as x>L the shock wave is downstream.

###### This question can be found in Fundamentals of Physics, 10th edition, chapter 2, question 12.

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Glad it helped! Best of luck in your studies.

So, just a thought/question, if people are still here.. – If we are trying to teach ourselves to solve problems such as these, what is a good strategy to be self-sufficient one day? Any problem-solving advice/strategy would be greatly appreciated. What is the expectation (time) that we won’t need to seek out solutions in the back of the book or google to solve? Thanks in advance. Merry Christmas

This is from my personal experience as a university student. The more questions you do, the better you get at it. In fact, I think at some point, after doing so many questions, the second you see a question, you start to formulate a method to solve it. It becomes a natural instinct. I guess one way to look at is the way we look at simple mathematics. When we first learned the Pythagorean theorem, it wasn’t simple, at least not the first day, but now, it’s almost second nature to us, and that’s simply because we had to solve many questions involving the same type of principles over and over again. Eventually, you also start getting an idea of whether the answer you got is right or wrong. It’s only a matter of how many questions it takes for us to reach that point. This is completely dependent on each of us. Some students gain insight faster than others, but all it means for others is that we just keep trying. Though at first, it seems like these questions can be daunting, they do become much easier as time goes on.

Some simple tips I can say is to make sure you list out what is given in a question. As simple as this seems, it’s incredible how helpful this is. Also, draw a diagram, always draw it out if you can, even if it’s just a simple line drawing. We can think better when we visualize things. Don’t skip the hard questions. Try them, figure out where you went wrong, and then do a similar question until you get it. Eventually, it gets much easier. Lastly, teach what you learn. You’ll find that when you try to teach something, you have to think twice as hard to explain it in simpler terms, and it always leads to you gaining a better understanding of the problem. If you don’t have anyone to teach, pretend to teach yourself, in the simplest way possible. It will do wonders!

Have a very Merry Christmas and a very happy new year! Best of luck with your studies ðŸ™‚