Designing a Speedometer
Suppose we want to design a speedometer for a high-speed train. Furthermore, suppose we already have a working clock and electronic odometer, and can use their output at any time and as often as we wish. How would you use the existing systems to design an electronic speedometer? Let's examine these questions.

  1. Suppose we have the following time and odometer readings:

    Time (t in minutes) 05 1015 202530
    Distance (d in kilometers) 01022 355064 78

    1. What was the average speed of the train during the first 30 minutes? Leave answers here and elsewhere in kilometers per minute.

      Average speed during first 30 minutes: _______km/min (For this questions only, answer also in miles per hour: ______mph. Also, if we want K×S to be speed in mph when S is speed in km/min, then the conversion constant is K=_______)

    2. What is the average speed during the first 20 minutes? __________.

    3. Give a definition of average speed which is both comprehensive (working in the most general context) and precise. You may use/define other terms to help in your definition.

  2. Towards understanding what information and what computations we need in order to design a speedometer, suppose that we wish to determine what the proper reading should be for our speedometer when t=10 minutes during the 30 minute trip portion described above.

    1. What is the average speed during the first 10 minutes? __________.

    2. Why is this not a very good estimate?

    3. How does your answer in (a) compare to the true answer? How do you know? And what (reasonable) assumptions are you making about the motion of the train here? Explain.

    4. What is the average speed during the time interval between t=5 and t=15? __________.

    5. What is the average speed during the time interval between t=5 and t=10? The time interval between t=10 and t=15? Is there an operation you can perform on these two numbers which you can intuitively justify as giving an approximation of the speed at time t=10? Perform this operation; what do you get? Does this always happen?

  3. To get a more accurate estimate of the speed when t=10, your partner in design suggests that more frequent readings be taken from the time/odometer unit by the prototype speedometer unit.

    1. Do you think this will help? If yes, explain how this must (or might) improve things; if no, explain why the suggestion will not yield better results.

    2. You now have the following additional readings:

      Time (t in minutes) 7 8 9 10 11 12 13
      Distance (d in kilometers) 13.6 15.8 18.8 22 25.2 28.5 31.8

      Compute the average speeds during the following intervals:

      The interval between t=5 and t=15: ____________________.

      The interval between t=7 and t=13: ____________________.

      The interval between t=8 and t=12: ____________________.

      The interval between t=9 and t=11: ____________________.

    3. How confident are you of these numbers (as estimates of the actual speed of the train when t=10)? If you knew the train speed was never decreasing during an interval, could you get an even more accurate estimate?
  4. Sketch the function d(t) below. Label the t and d axes, too, of course. Draw in also lines giving a geometrical representation of some of your answers above, and label them as such (Hint: average speed corresponds to the slope of a particular secant line).

    d  |
    i  |
    s  |
    t  |
    a  |
    n  |
    c  |
    e  |
    ____________________________________________________________ t-axis (minutes)
                                 t i m e
What did you learn about "velocity at an instant" (which is what a speedometer is supposed to give you, after all), and how it differs from, and how it relates to average velocity?

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