**Document ID**: 10_02_00_2

**Date Received**: 2000-10-02 **Date Revised**: 2002-11-20 **Date
Accepted**: 2002-11-29

**Curriculum Topic Benchmarks**: S12.4.7, S14.3.1, S14.3.2, S14.3.4,
S14.4.1, S15.3.1, S15.3.2, S15.3.3, M1.3.6, M1.3.8, M4.3.3, M4.3.20,
M4.4.3, M8.3.3

**Grade Level**: \[9-12\] High School

**Subject Keywords**: reaction time, experiments, gravity, acceleration,
checking answers

**Rating**: Advanced

# Right Place, Wrong Time

**By**: Murray F Spiegel, Telcordia Technologies, 445 South St, Room
1A-228R, Morristown NJ 07960 e-mail: spiegel@research.telcordia.com

**From**: The PUMAS Collection http://pumas.jpl.nasa.gov

©2002, California Institute of Technology. ALL RIGHTS RESERVED. Based on
U.S. Gov\'t sponsored research

It\'s important to check the results from an experiment. Does the result
make sense? Does it follow from other facts that are known? From the
standpoint of teaching High School science, checking if one\'s results
are sensible adds an additional layer of safety that the results are
correct. (From the standpoint of advancements in Science, a basic
research tenet is that results must be repeatable and not just a fluke.
Professional scientists were occasionally discredited when their data
can\'t be replicated; usually, they have made some blunder, but in rare
instances, it turns out some aspects of their data has been faked.)

The following story can be used to teach that results must be checked
against known facts to see if they\'re reasonable. It is designed to be
a follow up to the PUMAS example "The Fall of the Ruler." Some of the
background information in that example is repeated here for ease of
understanding.

**Human Reaction Time \[RT\] Background**

Human Reaction Time \[RT\] is defined as the time it takes a Human
(you!) to react to an event such as a light or a sound. Reaction Times
come into play during emergencies (\"Quick! A cow is on the tracks! Stop
the train!\") and more ordinary events, such as stopping at a stop
light, answering a ringing phone, turning to face someone who\'s talking
to you. \[For additional background, see the PUMAS example: "The Fall of
the Ruler."

You can measure human RTs using a simple ruler that has been calibrated
in milliseconds (msec), rather than inches or meters. Gravity causes
objects to drop towards Earth. The acceleration due to gravity is
incorporated in the equation relating distance to time:

D = 0.5 \* A \* T^2^

where A is approximately 10 meters / second^2^, T is time in seconds,
and D is distance in meters. Thus, a ruler drops this distance in 1
second:

D = 0.5 \* 10 \* (1)^2^ = 5 m (about 200 inches)

and in 0.1 second (100 msec), the distance in meters is

D = 0.5 \* 10 \* (0.1)^2^ = 0.5 \* 10 \* 0.01 = 0.05 meters (about 2
inches)

**Checking One\'s Results: Not Even School-Grade Materials are Safe**

When I was in graduate school, I helped my professor revise an
\"Introduction to Psychology\" class, in which students were taught how
to run and analyze experiments.

In a teaching-supplies catalog, the professor found a plastic ruler for
measuring human RT. The ruler eventually formed the basis for one of the
classes, essentially the second activity described in the PUMAS example
"The Fall of the Ruler". However, in testing the ruler, I repeatedly got
RTs that were *too* good. From classic experiments, we knew that human
RTs are typically indicated as 180 msec or longer, but these results
were sometimes as small as 130 msec. I was known as a quick person, but
getting an RT measurement 30% better than human capability was *very*
suspicious. We checked the equation relating time and distance and found
the ruler was wrong! By looking at the marks on the incorrect ruler, we
were able to figure out the manufacturer\'s error.

I was stopping the ruler after a 7 inch drop. Using the time-distance
equations above, that indicates an RT of about 190 msec. The incorrect
ruler marked 7 inches as a reaction time of 135 msec. We tried to
determine the manufacturer\'s error in two ways. They might have left
off the square of the time factor, giving an incorrect equation for
distance:

D = 0.5 \* A \* T

That would have left a linear relation between distance and time. But,
the marks on the ruler did get further apart for longer distances. (All
objects drop towards the ground faster with time - in one-tenth of a
second the ruler drops 2 inches, but in one second, it doesn\'t drop 10
times further, but 100 times more). Because the ruler had time marks
spaced by variable amounts, we suspected the manufacturer\'s equation
included the square of T.

Next we asked what would happen if the manufacturer had mistakenly
removed the factor of 0.5:

D = A \* T^2^

This equation puts the mark for 100 msecs (0.1 sec) around four inches:

D = 10 \* 0.1^2^ = 10 \* 0.01 = 0.1 meters (approximately 3.9 inches).

The values from this equation matched the marks on the defective ruler:
Four inches was marked as 100 msec, and \"my\" RT distance (7 inches)
was marked between 130-140 msec. Thus, we determined what a correctly
marked \"time ruler\" should have had, and we also discovered the error
in the equation used by the ruler maker. We were amazed that a teaching
device, manufactured in the thousands and sold nationally to schools,
would have such a fundamental error. Evidently, the manufacturer had not
checked their results.

**Suggested Activities**

As a follow up to activities in The Fall of the Ruler, you can:

1\) Construct a ruler based on the erroneous equation, and have students
collect RTs and show them to be better than published reports of human
RTs. Then ask students to see if they can identify the error responsible
for your defective ruler\'s markings.

Shorter activities using this example could be:

2\) Relate the story, asking students to try out different equations and
determine what the error was;

3\) Relate the story, asking students to compare time plots for a
\"good\" and \"defective\" time rulers;

**Lessons to learn**

1\) Whenever possible, check that your results are \"reasonable,\"
especially if you can find prior knowledge that is similar to your
result. In our case, we knew from carefully run prior experiments that
\"reasonable\" RTs couldn\'t be less than 150 msec.

2\) When using equations, it isn\'t enough to get the functional form
right, in this case: D=AT^2^. You also need to get the coefficients
right. The manufacturer\'s equation was missing the factor of 0.5, which
made the RT measurements off by the square root of 2.

Below is a table of the relationship between time (in msec) and
distance, both for a correct-time ruler and for the defective time
ruler. \[In this table, we\'ve used a more exact figure for gravity,
9.8 meters per second squared\]. You can see that both rulers would have
\"marks\" at about 11.1 inches. The correct ruler says that\'s an RT of
240 msec, whereas the incorrect one says it is a RT of 170 msec.
240/170 = 1.41, or the square root of 2.

+--------+------------+-------------+----------------+----------------+
| **     | **         | *           | **Wrong        | **Wrong        |
| Time** | Distance** | *Distance** | Distance**     | Distance**     |
|        |            |             | (meters)       | (inches)       |
| (msec) | (meters)   | (inches)    |                |                |
+--------+------------+-------------+----------------+----------------+
| 50     | 0.012      | 0.482       | 0.025          | 0.965          |
+--------+------------+-------------+----------------+----------------+
| 60     | 0.018      | 0.694       | 0.035          | 1.389          |
+--------+------------+-------------+----------------+----------------+
| 70     | 0.024      | 0.945       | 0.048          | 1.891          |
+--------+------------+-------------+----------------+----------------+
| 80     | 0.031      | 1.235       | 0.063          | 2.469          |
+--------+------------+-------------+----------------+----------------+
| 90     | 0.040      | 1.563       | 0.079          | 3.125          |
+--------+------------+-------------+----------------+----------------+
| 100    | 0.049      | 1.929       | 0.098          | 3.858          |
+--------+------------+-------------+----------------+----------------+
| 110    | 0.059      | 2.334       | 0.119          | 4.669          |
+--------+------------+-------------+----------------+----------------+
| 120    | 0.071      | 2.778       | 0.141          | 5.556          |
+--------+------------+-------------+----------------+----------------+
| 130    | 0.083      | 3.260       | 0.166          | 6.520          |
+--------+------------+-------------+----------------+----------------+
| 140    | 0.096      | 3.781       | 0.192          | 7.562          |
+--------+------------+-------------+----------------+----------------+
| 150    | 0.110      | 4.341       | 0.221          | 8.681          |
+--------+------------+-------------+----------------+----------------+
| 160    | 0.125      | 4.939       | 0.251          | 9.877          |
+--------+------------+-------------+----------------+----------------+
| 170    | 0.142      | 5.575       | 0.283          | 11.150         |
+--------+------------+-------------+----------------+----------------+
| 180    | 0.159      | 6.250       | 0.318          | 12.501         |
+--------+------------+-------------+----------------+----------------+
| 190    | 0.177      | 6.964       | 0.354          | 13.928         |
+--------+------------+-------------+----------------+----------------+
| 200    | 0.196      | 7.717       | 0.392          | 15.433         |
+--------+------------+-------------+----------------+----------------+
| 210    | 0.216      | 8.507       | 0.432          | 17.015         |
+--------+------------+-------------+----------------+----------------+
| 220    | 0.237      | 9.337       | 0.474          | 18.674         |
+--------+------------+-------------+----------------+----------------+
| 230    | 0.259      | 10.205      | 0.518          | 20.410         |
+--------+------------+-------------+----------------+----------------+
| 240    | 0.282      | 11.112      | 0.564          | 22.224         |
+--------+------------+-------------+----------------+----------------+
| 250    | 0.306      | 12.057      | 0.613          | 24.114         |
+--------+------------+-------------+----------------+----------------+

The author thanks Ron Menendez and Chuck Watson for their help in
preparing this example. The author's academic background is in
Experimental Psychology, having done fundamental research in how the
auditory system processes complex sounds. His proudest moment as a
graduate student wasn't getting an inhumanly small Reaction Time, but
testing a major symphony orchestra for its frequency perception. The
only prior study (75 years earlier!) of orchestra musicians was of the
Vienna Court Orchestra and included Gustav Mahler as a listener. The
author's current work involves speech technology for telecommunication
services.