Galileo didn't do the experiment exactly the same way we're going to do it. There was no such unit as seconds or centimeters at the time of his experiment. However he did have, and use, pendulums, even though he didn't know about the formula T = .2 sqrt(L) (which applies only to lengths in cm and periods in seconds).
First Experiment:
Every individual will do this trial independent of everyone else.
Start with one ramp on at a small slope:
Using a ramp, marble, pendulum and ruler determine the average velocity of the marble, starting from rest, and the slope of the ramp.
Assuming that the velocity changes at a uniform rate sketch a graph of velocity vs. clock time for your marble. Your graph should be a straight line segment starting from the origin, and the average velocity will occur at the midpoint of this segment.
Determine the final velocity of the marble.
Determine the average rate of change of the velocity of the marble with respect to clock time.
Find the slope of your ramp.
Write down the ordered pair (sramp lope, ave rate of change of velocity).
Repeat this for a second slope which is about double the first.
Plot your two ordered pairs on a graph of ave. rate of change of velocity vs. ramp slope. Find the slope of your graph between these two points.
Second Experiment (in your Group of 5):
Elect, appoint or coerce a different tyrant and a different recorder.
Your group will also need a ramp measurer, a pendulum holder and an observer.
The ramp measurer will set up the first ramp and measure the information you will need to calculate its slope and length. The tyrant should make sure that the measurements make sense; before the recorder writes down the results that person should question the results if they don't seem right or reasonable.
Using repeated runs of the marble down the ramp determine as accurately as possible a pendulum length which will allow the pendulum to start at the instant of release end one of its half-cycles at the instant the pendulum reaches the end of the ramp.
Take the data necessary to determine the same quantities requested in the first experiment. Do not do the calculations at this point--the calculations are for homework.
Repeat the process for at least four ramp slopes, with no slope very steep and
with each slope significantly different than the others.
For homework repeat the analysis of the first trial, obtaining at the end a graph of average rate of change of velocity vs. ramp slope.
Fit your best straight line to this graph and determine the slope of that straight line.