Linear Motion & Galileo
The following is adapted with permission from the University of Dallas Fall ‘01 General Physics I
Laboratory Manual, Adaptation: John Boehringer
When Galileo introduced the concept of uniform acceleration, he defined it as equal
increases in speed in equal intervals of time. This experiment is similar to the one
discussed by Galileo in his book, Dialogues Concerning Two New Sciences, in which
he assumed that a ball rolling down an incline accelerates uniformly. Rather than
using a water clock, as Galileo did, you will use a motion detector and computer.
This will enable you to obtain very accurate measurements with a high degree of
precision. From these measurements you should be able to decide for yourself
whether Galileo’s assumption was valid or not.
Perhaps more significantly, indeed counter to majority thinking at the time that lingers
as common misconception today, Galileo further argued that objects of different sizes
and weights would accelerate at the same rate under the influence of gravity. Of
course, the idea of gravity as Newton would codify it did not exist at the time, and
Galileo was careful to place his observations in the content of his experiment alone.
We now know that there were many much broader implications to his work.
Speed was very difficult for Galileo to measure. He used instead two quantities that
were significantly easier to obtain: total distance traveled and elapsed time. In this
experiment your use of a sonic motion probe will eliminate this barrier, allowing you
to calculate instantaneous speed at many points down the incline. The data that you
acquire in one trial will be far greater that Galileo was able to obtain over many, many
successful experiments.
Problem
 How can you determine the acceleration of a cart moving down an inclined
plane?
 How do you know if that acceleration is constant?
 What kind of algebraic functions describe the velocity as a function of time and
acceleration as a function of time?
Materials
Windows PC
Lab Pro Interface
Vernier LoggerPro software
Vernier Motion Detector
Pasco Dynamics bench w/ cart (or ball)
Preliminary Questions (Answer these BEFORE you begin.)
1. List and explain (if necessary) some observations that led people of Galileo’s time
to believe that heavier objects fall faster than lighter objects.
2. Drop a small ball and a large ball from the same height at the same time. Did that
large one hit first, last or at the same time? If you assume both balls were the same
mass, which one should hit first and why?
3. What would happen if you again simultaneously dropped two identical balls, but
this time held one ball about 30 cm above the other, releasing them at the same time.
Would the distance between the balls increase, decrease or remain the same?
Why? Since the fall is short you can see why people in earlier times had difficulty
with this and other questions of motion.
Procedure
1. Connect the motion detector to the LabPro interface and the interface to the PC.
2. Place the motion detector on its long side, pointed down a 1 to 3 m long incline. The
incline should be between 5 and 10 degrees with the horizontal. Usually one text book is
enough to prop up the ramp sufficiently.
3. Prepare the computer for data collection by opening “Exp03” from the Physics with
Computers experiment files Logger Pro. Two graphs should appear on the screen. On the
top graph the vertical axis has distance scaled from 0 to 3 meters. The second graph has
velocity scaled from 0 to 2 m/s. The horizontal axis of both graphs is the same time interval
from 0 to 3 s.
4. Position a dynamics cart (or ball) about 0.4 m down the incline (remember the motion
detector cannot sense objects much closer than this).
5. Click Collect to begin data collection. Release the cart when you hear the sensor begin to
click.
6. Repeat the procedure if you are not satisfied with your results. Print or accurately (i.e.,
with a ruler) sketch your graphs of distance versus time.
7. Click on the velocity vs. time graph. Click on the Examine tool on the toolbar. Move the
cursur to a point about one-fourth of the way down the incline. Record, in the data table
below, the value of time and velocity at that point. Starting from that point, record time and
velocity data for every 0.2 seconds until you have recorded ten points or until you read the
point that corresponds with the end of the acceleration.
Data Point
1
2
3
4
5
6
7
8
9
10
Slope of v-t graph
Average
acceleration
Time
(s)
Data Table
Speed
(m/s)
Change in speed
(m/s2)
Drawing Conclusions
1. Calculate the change in speed between each of the points in your data table
above. Enter these values in the right column of your data table.
2. Galileo’s definition of uniform acceleration is equal increases in speed in equal
intervals of time. Does your data support of refute this definition for the motion of an
object on an incline? Explain.
3. Was Galileo’s assumption of constant acceleration down an incline valid? How
does your data support this conclusion?
4. Calculate the average acceleration of the ball between the first and last time
recorded (t1 and t10) using your data and the definition of average acceleration:
v v final  vinitial
a

t t final  tinitial
Record in your table.
Making a Mathematical Model
1. Does the distance versus time graph appear to be a simple algebraic curve? Which curve
or line? Try fitting various functions to the portion of the data corresponding to the free rolling
by dragging across that time interval and clicking the Curve Fit button. Select a function by
scrolling the list and click Try Fit to replot and see how well the function fits the data. Choose
the most simple function that fits best (there may be several, take the lowest order function)
and click O.K. to place that fit in the main graph window. Record both the equation and
parameters of the fitted equation. Print or accurately sketch the graph. Include the curve fit
and parameters.
(Write your equation and parameters HERE)
2. Similarly, does the velocity versus time graph appear to follow a simple algebraic curve?
Using the process above, choose the simplest function that fits the data well and record the
parameters of the fitted equation. Print or sketch the graph with the curve fit information.
(Write your equation and parameters HERE)
3. Look at the curve fit equation for the distance versus time graph. How does the constant
coefficient of the t2 term in this curve relate to the slope of the velocity versus time graph?
4. Now Look at the curve fit equation for the velocity versus time graph. Does the fitted
function have a constant slope? What does that slope mean? What are its units? Record in
the data table.
5. Recall Newton’s kinematics formula x(t) = x0 + v0t + ½at2. Consider your answers to
numbers 3 & 4, above. Do your coefficients reflect this same relationship?
Assessing
1. Suppose you were to repeat the experiment many times and each time you elevated the
track even more. Would you expect the values you measure for average acceleration to
increase, decrease or remain the same and why?
2. What value would the average acceleration approach as the angle of the incline neared
90 degrees?
3. Two identical carts, A and B, roll down the same track. Cart A starts slightly higher up the
track than B. The carts are released at the same instant. Would you expect cart A to catch
up to cart B? Explain your reasoning and include support from your lab.
Two identical carts, C and D, roll down identical tracks that are inclined by the same amount.
The carts start at the same position on the tracks. Cart C is released first, then one-half
second later cart D is released.
4. Which cart would you expect to have greater acceleration down the track? Why?
5. Which cart would you expect to reach the bottom of the track first? Why?
6. Do you think the distance between the carts as they roll down the incline in this new
scenario will increase, decrease or remain the same? Try it if you are uncertain. Explain
what you see.
Extending
Start the cart at the bottom of the incline and roll it upwards gently, let it stop and roll back
down. Use the sensor. Compare the up and down motions of the cart. Do your conclusions
established in the lab still hold? Explain.