Lab
Galileo Ramps
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Statue of Galileo outside of the Uffizi, Florence
image courtesy of
Open Parachute
Galileo Galilei (1564-1642) noticed that objects experiencing uniformly accelerated motion produced an interesting pattern relating the number of time intervals through which they moves and the distance which they ultimately travelled.
image courtesy of
San Diego State University, Department of Natural Sciences
The spacing between locations shown in the diagram above could be charted as
time
position
change in position
0
0
1
1
1
= 1-0
2
4
3
= 4-1
3
9
5
= 9-4
4
16
7
= 16-9
5
25
9
= 25-16
Notice that the displacement between consecutive positions increased by odd integers. Also, notice that the total distance travelled, or net displacement since the objects were moving in a straight-line, from t = 0 was proportional to
t
^{2}
.
Although his time was not our present day definition of a second (he used his pulse, a water clock, and a pendulum) he was the first to define the concept of uniformly accelerated motion as
D
v/
D
t equaling a constant value. He had to test his definition indirectly since he could not measure an object's instantaneous velocity. He conducted his trials using an inclined plane on which bells were spaced at increasing odd-integer intervals. The bells assisted with his timing by providing an auditory alert of when his "
hard, smooth, and very round bronze ball
" arrived at each position on the ramp. The incline's enormous length (12 cubits, or roughly 5.5 meters) allowed him enough time to take accurate measurements. In the picture below, you can see a replica of Galileo's ramp.
image courtesy of
Arbor Scientific
We know from our discussions, that average velocity is defined as the quotient of net displacement over total time.
We also know from Galileo's definition that when an object experiences uniform acceleration,
and that its average velocity over an interval of time equals the average of its initial and final velocities during that time interval.
Using these facts, along with the requirement that the ball be released from rest, we have the equations,
Thus, Galileo devised a method of measuring accelerations. Although he never changed his ramp's elevation, we realize that different angles of elevation will result in different values for
D
t
and different values for
a
using the same displacement, or length of ramp.
Our experiment
In our experiment, we will model Galileo's ramp and his determination of the behavior for naturally occurring accelerated motion by using an inclined plane, a motion detector, and the LabPro software. As we conduct our experiment, we will gradually increase the incline's angle of elevation recording the height between the top of the table and the BOTTOM of the board with each set of trials.
Notice from the diagram above, that the sine of the incline's angle of inclination equals height/length, or h/L. This relationship will be important when we construct our graph.
What is the total length of your board in centimeters?
Starting with an initial incline elevated close to 10 cm, release your cart from rest a minimum of 40 cm in front of the motion detector, allow it to roll completely down the incline recording its motion on Lab Pro. Please CATCH the cart when it reaches the end of the incline, do NOT allow it to crash to the floor! As you watch the cart's position-time graph "grow" you will see the cart's accelerated motion (a parabola). As you watch its velocity-time graph "grow" you will see a linear relationship during the same time interval.
Highlight the appropriate "linear" section of the velocity graph (which corresponds with the parabolic section of the position graph) and fit a regression line. Record its slope in your data table. Note from the slope's units, that it represents the cart's acceleration (m/sec/sec). Gather information for a total of three trials at this angle.
Now elevate your board by at least 5 cm, recording your new height, and release the cart from rest three more times. Remembering with each trial, to record the slope of the regression line for the velocity-time graph displayed on LabPro. You will do this for a total of 7 different heights.
height
trial 1
trial 2
trial 3
average slope
sin
q
elevation
(cm)
(m/sec
^{2}
)
(m/sec
^{2}
)
(m/sec
^{2}
)
(m/sec
^{2}
)
h/L
1
2
3
4
5
6
7
Data Analysis
You will now use EXCEL to plot a scatter plot of average acceleration vs sin(
q
). Save your file with the format LastnameLastnameLastnameRamps.xls and email it to Mrs. Colwell.
List the members in your group in alphabetical order, last name first.
What was the slope of the trend line of your group's EXCEL graph?
What was the y-intercept of the trend line of your group's EXCEL graph?
What was the correlation coefficient of the trend line of your group's EXCEL graph?
Using the equation of your group's trend line, extrapolate the ball's average acceleration when
q
= 90º.
Conclusions
Did your data for the cart's average acceleration "fit" better when the angles were smaller at the beginning, or when they were larger at the end? Explain your choice.
If we could assume that the cart only translated or "simply slid" down the incline, your extrapolated acceleration value should be close to that of the acceleration due to gravity, or 9.8 m/sec
^{2}
. Calculate a percent error for your extrapolated value against this accepted value for the acceleration of a freely falling body.
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Resource Lesson:
RL -
Accelerated Motion: A Data Analysis Approach
RL -
Accelerated Motion: Velocity-Time Graphs
RL -
Analyzing SVA Graph Combinations
RL -
Average Velocity - A Calculus Approach
RL -
Chase Problems
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Chase Problems: Projectiles
RL -
Comparing Constant Velocity Graphs of Position-Time & Velocity-Time
RL -
Constant Velocity: Position-Time Graphs
RL -
Constant Velocity: Velocity-Time Graphs
RL -
Derivation of the Kinematics Equations for Uniformly Accelerated Motion
RL -
Derivatives: Instantaneous vs Average Velocities
RL -
Directions: Flash Cards
RL -
Freefall: Horizontally Released Projectiles (2D-Motion)
RL -
Freefall: Projectiles in 1-Dimension
RL -
Freefall: Projectiles Released at an Angle (2D-Motion)
RL -
Monkey and the Hunter
RL -
Summary: Graph Shapes for Constant Velocity
RL -
Summary: Graph Shapes for Uniformly Accelerated Motion
RL -
SVA: Slopes and Area Relationships
RL -
Vector Resultants: Average Velocity
Review:
REV -
Test #1: APC Review Sheet
Worksheet:
APP -
Hackensack
APP -
The Baseball Game
APP -
The Big Mac
APP -
The Cemetary
APP -
The Golf Game
APP -
The Spring Phling
CP -
2D Projectiles
CP -
Dropped From Rest
CP -
Freefall
CP -
Non-Accelerated and Accelerated Motion
CP -
Tossed Ball
CP -
Up and Down
NT -
Average Speed
NT -
Back-and-Forth
NT -
Crosswinds
NT -
Headwinds
NT -
Monkey Shooter
NT -
Pendulum
NT -
Projectile
WS -
Accelerated Motion: Analyzing Velocity-Time Graphs
WS -
Accelerated Motion: Graph Shape Patterns
WS -
Accelerated Motion: Practice with Data Analysis
WS -
Advanced Properties of Freely Falling Bodies #1
WS -
Advanced Properties of Freely Falling Bodies #2
WS -
Advanced Properties of Freely Falling Bodies #3
WS -
Average Speed and Average Velocity
WS -
Average Speed Drill
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Chase Problems #1
WS -
Chase Problems #2
WS -
Chase Problems: Projectiles
WS -
Combining Kinematics and Dynamics
WS -
Constant Velocity: Converting Position and Velocity Graphs
WS -
Constant Velocity: Position-Time Graphs #1
WS -
Constant Velocity: Position-Time Graphs #2
WS -
Constant Velocity: Position-Time Graphs #3
WS -
Constant Velocity: Velocity-Time Graphs #1
WS -
Constant Velocity: Velocity-Time Graphs #2
WS -
Constant Velocity: Velocity-Time Graphs #3
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Converting s-t and v-t Graphs
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Energy Methods: More Practice with Projectiles
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Energy Methods: Projectiles
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Force vs Displacement Graphs
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Freefall #1
WS -
Freefall #2
WS -
Freefall #3
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Freefall #3 (Honors)
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Horizontally Released Projectiles #1
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Horizontally Released Projectiles #2
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Kinematics Along With Work/Energy
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Kinematics Equations #1
WS -
Kinematics Equations #2
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Kinematics Equations #3: A Stop Light Story
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Lab Discussion: Gravitational Field Strength and the Acceleration Due to Gravity
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Position-Time Graph "Story" Combinations
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Projectiles Released at an Angle
WS -
Rotational Kinetic Energy
WS -
SVA Relationships #1
WS -
SVA Relationships #2
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SVA Relationships #3
WS -
SVA Relationships #4
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SVA Relationships #5
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Work and Energy Practice: An Assortment of Situations
TB -
2A: Introduction to Motion
TB -
2B: Average Speed and Average Velocity
TB -
Antiderivatives and Kinematics Functions
TB -
Honors: Average Speed/Velocity
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Kinematics Derivatives
TB -
Projectile Summary
TB -
Projectile Summary
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Projectiles Mixed (Vertical and Horizontal Release)
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Projectiles Released at an Angle
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Set 3A: Projectiles
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