Expert: Daniel Mazur - 4/26/2009

Question
QUESTION: hi
i working in a construction company.. building metro.. my boss give me a timetable diagram.. cant find the equation of it... a train at 35 time is in the distance of 8614 meter and after 15 minutes(means 50 time)it reaches to the distance 6116 meter and if the train has 200 m/min average speed, so what is the distance-time equation of it? i know for constant speed the equation is x=vt+x0 but it doesnt work for those digits.. would u help me? i need it quick.. at work now.. sorry for my bad english..
thanks


ANSWER: Hi Atusa,

I apologize for not receiving your question quickly enough, you may already have the answer. Your given values are:
t0 = 35 min
t1 = 50 min
x0 = x(t0) = 8614 m
x1 = x(t1) = 6116 m
vavg = 200 m/min

You are correct that "it doesn't work for those digits", because there is no "unknown" left in your equation (which is correct, by the way) and the numbers are not consistent. I suggest that you ask details about how solid is the "average speed" vavg. Averages are usually calculated between two stops or between terminals, but that doesn't mean that at every point of the way the train speed is the average speed.

So, what is the  "15-minute average speed" given by your numbers without vavg? It is
vavg15min = abs((x1-x0)/(t1-t0)) = 167 m/min. The abs() just takes care of a "minus", which is unimportant for your problem.

This 167m/min is not terribly far off the stated average speed 200m/min. I recommend that you ask the competent personnel, how EXACTLY was the average speed vavg measured. Different methods can make a lot of difference in results, even in averaging. For example, if your 15 minutes included waiting at stops and the other method didn't, that can be the source of the discrepancy. Like I said, your equation is correct, but you must decide (based on some good reason), which "average speed" value you should use. If you write me some details, I might be able to help you with the decision making.

Have a nice weekend!
Daniel

---------- FOLLOW-UP ----------

QUESTION: hi Daniel
thanks for replying to my question..it was my boss fault.. cuz he dint tell me maybe the digits r not real... so we still on average speed.. with an average speed of 200 m/min and having 3 trains and a 5000 meter distance, where should i install the switches.. where the trains may smash! at what distance? and why when i use 4 trains i need 2 switches?
thank u

Answer
Hello Atusa,

I will need you to tell me a bit more about the background of your problem, because I am a physicist and the best person you'd need for this is a civil/railway engineer.

Please verify that my understanding is correct. The distance L=5000m is the total length of the metro line, correct? Now, placing N trains on that line equidistantly may be done in two ways: a) both ends (marks 0m and 5000m) are occupied by train so that the trains split the line into (N-1) sections of equal length or b) only mark 0m is occupied by a train and the line is split by the trains into N sections. In case a) the train on mark 5000m cannot move any further, while in case b) all the trains can move on. Which case is that you are proposing? The average distances between trains (VERY roughly speaking) will be a) D=L/(N-1) and b) D=L/N.

In the second step you probably should consider the length of the trains LT. In my hometown the trains are about 100m long each, and for that you need to adjust the positions of the switches. Otherwise your train in the back may start braking too late and still smash into the rear of the train in front.

Lastly, do you really want to use only as few switches as possible? You see, if it was me, I would take the maximum speed (not average) and estimate the distance DS for a fully loaded train to come to full stop. Then I would simply put the switches in the intervals equal to DS+LT. Then the number of switches necessary will be NSW = 1 + L/(DS+LT) and the maximum number of train to safely use on the line will be Nmax = L/(DS+LT). Safety measures cannot be properly calculated based on average speed, you must use the maximum speed - the length of braking distance is proportional to v^2, and if v>vavg, you will be in trouble.

Please let me know those details I mentioned. If you really have a fixed number of trains and want to minimize the number of switches, tell me and I will elaborate on that for you.

Cheers,
Daniel