Mathematics 161                                  Laboratory 13                             December 3, 1992
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994

How Far to Harvard Square?

This is a two week laboratory.  Your final report is due Friday, December 11.  At the end lab today, you and your lab partner should give me an oral report of your progress.

Part 0:  The Data

Several MIT students rode the Boston subway from Kendall Square to Harvard Square, with a stop at Central Square. Using an accelerometer and a stopwatch, every second they measured and recorded the acceleration of the car in miles per hour per second.  The data they collected are listed below.  Notice that there are 231 pairs of data.

The data on acceleration of the subway car have been collected in the notebook file lab13.ma which is in the usual place.  If you choose not to use the electronic form of this notebook, enter the following command:

<<i:\course\root\math161\lab13.m

to assign the name data to the table of data shown on the preceding page.

Part 1:	The Assignment

Assuming that the subway car started from rest, use these data to

1)	determine v(t), the velocity of the car, at the times
     t = 0, 1, ... , 230, and make a plot of v(t);

2)	determine the maximum velocity (in miles per hour) achieved by the car and the
     time at which that maximum velocity occurred;

3)	determine the distance (in miles along the subway track) from Kendall Square to
     Harvard Square.

Part 2:	Getting Started

You can make a plot of the acceleration data using

ListPlot[data, PlotJoined->True]

(The command ListPlot[list], where list is a list of ordered pairs {x, y}, causes the points (x, y) to be plotted.  The option PlotJoined -> True causes the points which are plotted to be connected by line segments.)

In solving the problems listed above, you will need to get your hands on individual points in the list data.  Suppose you want to check the acceleration at time t = 100.  Since the first point in the list data corresponds to the time t = 0, you are actually after the 101st data pair in the list.  To get this pair, type

data[[101]]

If you want only the acceleration at time t = 100, you can use

data[[101]][[2]]

to pull off the second coordinate in the 101st ordered pair.  We can use this idea to define an acceleration function, a(t).

a[t_] := data[[t+1]][[2]]

Notice that the variable t can take only the integer values 0, 1, ... , 230.

With the function a(t) defined, we can plot our acceleration with the command

ListPlot[Table[{t,a[t]},{t,0,230}], PlotJoined->True]

(Notice that Table[{t,a[t]},{t,0,230}] simply generates the list of ordered pairs that we called data to begin with.)

Part 4: 	Some Things to Think About

After you have made the various plots and definitions described above and gotten a good feeling for the data, you might find it useful to discuss the following questions with your lab partner.

1)	How can you compute the (approximate) velocity from the acceleration data which
    you have been given?

2)	How will you make the required plot of the velocity?

3)	When you have made your plot of the velocity, what can you look for to get an idea
    of whether or not your computations are correct?

4)	If you knew the position function of the subway car, you could obtain the answer of
    the third problem.  (Couldn't you?)  How can you compute the position from the
    velocity?  Do you really need to know the position of the car at all times in order to
    solve the third problem?

Write an essay in which you give the solutions of the three problems you have been asked to address and explain how you obtained your solutions.  (One page may do the job; you will need two pages at most.)  Notice that your report should include a printout of the graph of the velocity of the subway car. You may or may not want to include other printouts.  Be sure to examine your results critically.  Try to explain any discrepancies between your results and what you might have expected to see.

data=
{{0, 0.1}, {1, 0.6}, {2, 0.9}, {3, 1.6}, {4, 1.8}, {5, 1.9}, {6, 2.}, {7, 2.},
{8, 2.}, {9, 2.}, {10, 2.}, {11, 1.9}, {12, 1.9}, {13, 1.8}, {14, 1.7},
{15, 1.6}, {16, 1.5}, {17, 1.4}, {18, 1.2}, {19, 1.1}, {20, 1.}, {21, 0.9},
{22, 0.8}, {23, 0.7}, {24, 0.6}, {25, 0.6}, {26, 0.5}, {27, 0.4}, {28, 0.4},
{29, 0.3}, {30, 0.3},{31, 0.2}, {32, 0.2}, {33, 0.2}, {34, 0.2}, {35, 0.2},
{36, 0.2}, {37, 0.2}, {38, 0.2}, {39, 0.2}, {40, 0.2}, {41, 0.2}, {42, 0.2},
{43, 0.3}, {44, 0.3}, {45, 0.3}, {46, 0.3}, {47, 0.3}, {48, 0.4}, {49, 0.4},
{50, 0.4}, {51, 0.3}, {52, 0.3}, {53, 0.3}, {54, 0.2}, {55, 0.2}, {56, 0.1},
{57, 0.1}, {58, 0.}, {59, -0.1}, {60, -0.1}, {61, -0.2}, {62, -0.3}, {63, -0.3},
{64, -0.4}, {65, -0.4}, {66, -0.4}, {67, -0.5}, {68, -0.4}, {69, -0.4},
{70, -0.4}, {71, -0.4}, {72, -0.3}, {73, -0.3}, {74, -0.2}, {75, -0.2},
{76, -0.1}, {77, -0.1}, {78, -0.1}, {79, 0.}, {80, -0.1}, {81,-0.1}, {82, -0.2},
{83, -0.2}, {84, -0.4}, {85, -0.5}, {86, -0.6}, {87, -0.8}, {88, -1.},
{89, -1.2}, {90, -1.4}, {91, -1.6}, {92, -1.8}, {93, -2.}, {94, -2.1},
{95, -2.2}, {96, -2.3}, {97, -2.4}, {98, -2.5}, {99, -2.4}, {100, -2.3},
{101, -2.2}, {102, -2.}, {103, -1.9}, {104, -1.6}, {105, -1.1}, {106, -0.8},
{107, -0.5}, {108, -0.2}, {109, 0.}, {110, 0.}, {111, 0.}, {112, 0.}, {113, 0.},
{114, 0.}, {115, 0.}, {116, 0.}, {117, 0.}, {118, 0.2}, {119, 0.5}, {120, 1.1},
{121, 1.7}, {122, 2.2}, {123, 2.2}, {124, 2.2}, {125, 2.2}, {126, 1.9},
{127, 1.7}, {128, 1.5}, {129, 1.3}, {130, 1.5}, {131, 1.7}, {132, 2.}, {133, 2.},
{134, 2.1}, {135, 2.}, {136, 2.}, {137, 1.8}, {138, 1.3}, {139, 1.4},
{140, 1.1}, {141, 0.8}, {142, 0.6}, {143, 0.4}, {144, 0.1}, {145, 0.1},
{146, 0.2}, {147, 0.3}, {148, 0.3}, {149, 0.3}, {150, 0.4}, {151, 0.4},
{152, 0.5}, {153, 0.2}, {154, 0.1}, {155, 0.2}, {156, 0.2}, {157, 0.2},
{158, 0.2}, {159, 0.2}, {160, 0.4}, {161, 0.6}, {162, 0.8}, {163, 0.6},
{164, 0.4}, {165, 0.2}, {166, 0.}, {167, 0.}, {168, 0.}, {169, 0.}, {170, 0.},
{171, 0.}, {172, 0.}, {173, 0.}, {174, 0.1}, {175, 0.1}, {176, 0.1}, {177, 0.1},
{178, 0.3}, {179, 0.2}, {180, 0.1}, {181, 0.2}, {182, 0.2}, {183, -0.8},
{184, -0.8}, {185, -0.5}, {186, -0.5}, {187, -0.5}, {188, -0.5}, {189, -0.5},
{190, -0.4}, {191, -0.4}, {192, -0.4}, {193, -0.4}, {194, -0.4}, {195, -0.3},
{196, -0.3}, {197, -0.3}, {198, -0.3}, {199, -0.4}, {200, -0.8}, {201, -1.5},
{202, -1.6}, {203, -1.7}, {204, -1.8}, {205, -0.9}, {206, -1.1}, {207, -1.2},
{208, -1.4}, {209, -1.5}, {210, -1.8}, {211, -1.8}, {212, -1.6}, {213, -1.4},
{214, -1.2}, {215, -1.}, {216, -1.2}, {217, -1.8}, {218, -2.2}, {219, -2.1},
{220, -2.}, {221, -1.9}, {222, -1.7}, {223, -1.7}, {224, -1.5}, {225, -1.},
{226, -0.7}, {227, -0.3}, {228, -0.1}, {229, 0.}, {230, 0.}};

TableForm[Table[Table[
{data[[33j+i,1]],data[[33j+i,2]]},{j,0,6}],
{i,33}],TableDirections->{Column,Row,Row}]