Attal 1
Kush Attal
Mrs. Eichenberg
Period 4
27 September 2015
Honors Physics: Lab to Determine the Acceleration of Free Fall – 2015/16
Abstract
The purpose of this experiment is to find the acceleration of free fall mainly due to gravity, and determine
the relationships between acceleration, velocity, and distance. The hypothesis is that the acceleration of
the photogate in free fall will be 9.8m/ s2with a ±0.2 for precision (always in the negative direction or
towards the earth unless otherwise stated) because the only force working on the object is gravity, a
known force calculated by Cavendish. In addition, the hypothesis also states that as acceleration continues
due to greater distance, velocity will increase in the same direction (all velocity measurements will be
taken with their direction as toward the earth). When testing for repeatability and precision, the results of
the experiment indicate an average acceleration of 9.7095 m/ s2, a fairly accurate measurement with only
0.92% percent error from the accepted value of gravity (9.8 m/ s2) in Part 1. When testing for velocity vs.
acceleration, the results in Part 2 indicate a greater velocity in successive trials where the distance
between start and finish was greater. For example, while Trial 1 had a velocity of 0.5838 m/s, Trial 4
contained a velocity of 2.4037 m/s, nearly 4 times the velocity of Trial 1 only from an additional 30 cm.
Therefore, the hypothesis of the experiment was proven by the results of this experiment.
Background
An object is in free fall when the only force acting on it is the earth’s gravitational force. No other forces
can be acting; in particular, air resistance must be either absent or so small as to be ignored. When the
object in free fall is near the surface of the earth, the gravitational force on it is nearly constant. As a
result, an object in free fall accelerates downward at a constant rate. This acceleration is usually
represented with the symbol g.
In this experiment uses a very precise timer connected to the computer and a Photogate. The Photogate
has a beam of infrared light that travels from one side to the other. It can detect whenever this beam is
blocked. Students will drop a piece of clear plastic with evenly spaced black bars on it, called a Picket
Fence. As the Picket Fence passes through the Photogate, the computer will measure the time from the
leading edge of one bar blocking the beam until the leading edge of the next bar blocks the beam. This
timing continues as all eight bars pass through the Photogate. From these measured times, the program
Logger Pro will calculate the velocities and accelerations for this motion and graphs will be plotted.
Two of the relationships between distance, time and velocity as given in the Kinematics unit are:
v = vo + at
where a = g (acceleration due to gravity)
d = vot + ½ at2
Materials
Ring stand
Clamp
Vernier Photogate
Vernier Lab Pro
Computer with Vernier Logger Pro software
Vernier Picket Fence (measure and state dark/clear intervals and report it in your final report HERE)
Attal 2
Meter stick (Precision: .5 mm)
Something to put on floor to pad the landing of the picket fence so it doesn’t break (someone’s jacket,
etc.)
Diagram
Ring Stand
Computer with Vernier Logger Pro
Photogate
Figure 1: Set-up
Towel to break fall
Safety Precaution(s):
-don’t play with or break any equipment
Procedure / Data Collection
1. Set up the experimental apparatus. Three items connect into the Lab Pro. The photogate plugs into
the DIG/SONIC1 port of the Lab Pro (white BTA plug), see Figure 1 below. The Lab Pro has a cable
that plugs into its port that is labeled with a network symbol and the other end of that cable is a USB
connector that plugs into a USB port on the computer, see Figure 2. The third item that plugs into the
Lab Pro is the AC adaptor, which then plugs into a wall socket. When those three things have been
connected, the screen on Logger Pro should show the Photogate as connected to DIG/SONIC1 as
shown in Figure 1.
Figure 2: Lab Pro screen that appears on computer on Logger Pro 3.8.7
(this graphic shows two photogates, but this lab only uses one)
Figure 3: Lab Pro cable
2. Assemble the ring stand and photogate so the picket fence is free to fall through it to land on something soft
below, as shown in Figure 3. Use a student’s hand to block and unblock the photogate beam of infrared light
and verify on the Logger Pro status bar that the photogate state switches as the hand blocks and unblocks the
photogate. This is to verify that the photogate is functional before beginning experiments.
Attal 3
Picket
fence
Figure 4: Photogate Free Fall Demonstration
3. PART 1: REPEATABILITY
To collect data, click on the green
button at the top of the screen. When dropping the picket
fence, the objective is to drop it straight down through the photogate, so it doesn’t come in contact
with anything, which would interfere with the data collection. Set the picket fence at a height just
barely above the photogate and drop it through the photogate, being careful to release it without
giving it any initial velocity (push/pull). To stop data collection, click on the
button at the
top of the screen. To save the trial’s data, click on Experiment/Store Latest Run. Repeat this
experiment a total of 3 times (2 more times) and transfer that data to Excel. To transfer the data, click
on File/Export and choose “CSV” file type to export, then save the file in the Physics folder on the
desktop with a name that is unique to your group. This file then will be opened in Excel, either in
class or at home (bring flash drive to transport file). After opening the file in Excel, save it as an
Excel file to change the file type from CSV to Excel.
4. PART 2: VARYING HEIGHTS
From four different (measured and documented) heights that the team chooses, drop the picket fence and
allow it to free fall, and measure the distance, the velocity and acceleration due to gravity for each of those
intervals. After each run be sure to “Store Latest Run” so the data is saved. After the fourth trial, export the
data to Excel as directed in step 3 above.
Table 1 – Part 1, Raw Data
Run 1: Time (s)
Elapsed Time(s)
0.322419
0
0.377902
0.055483
0.414888
0.092469
0.444704
0.122285
0.470399
0.14798
0.493308
0.170889
0.514183
0.191764
Run 1: Distance
(m)
0
0.05
0.1
0.15
0.2
0.25
0.3
Run 1: Velocity
(m/s)
1.172
1.532
1.821
2.071
2.294
Run 1: Acceleration
(m/s²)
9.75
9.73
9.69
9.74
9.71
Attal 4
Run 2: Distance
Run 2: Velocity Run 2: Acceleration
Run 2: Time (s)
Elapsed Time (s)
(m)
(m/s)
(m/s²)
0.881927
0
0
0.933383
0.051456
0.05
1.222
9.74
0.969183
0.087256
0.1
1.57
9.7
0.998392
0.116465
0.15
1.852
9.63
1.023716
0.141789
0.2
2.097
9.64
1.046383
0.164456
0.25
2.315
9.67
1.067083
0.185156
0.3
Run 3: Time (s)
Elapsed Time (s)
0.575083
0
0.626782
0.051699
0.662612
0.087529
0.691821
0.116738
0.717094
0.142011
0.739712
0.164629
Run 3: Distance
(m)
Run 3: Velocity
(m/s)
0
0.05
0.1
0.15
0.2
0.25
Run 3: Acceleration
(m/s²)
1.22
1.57
1.855
2.101
2.319
9.79
9.73
9.79
9.7
9.61
Table 2- Part 2, Raw Data:
Run 1:
Time (s)
Elapsed
Time (s)
1.045091
1.103025
1.140525
1.170627
1.196419
1.21941
1.240384
0
0.057934
0.095434
0.125536
0.151328
0.174319
0.195293
Run 1:
Run 1:
Run 1:
Run 2:
Distance Velocity Acceleration Time (s)
(m)
(m/s)
(m/s²)
0
0.05
0.1
0.15
0.2
0.25
0.3
1.149
1.515
1.811
2.063
2.284
9.86
9.69
9.93
9.68
9.51
0.781721
0.814216
0.841584
0.865619
0.887308
0.907284
0.925884
Elapsed
Time (s)
0
0.032495
0.059863
0.083898
0.105587
0.125563
0.144163
Run 2:
Run 2:
Run 2:
Distance Velocity Acceleration
(m)
(m/s)
(m/s²)
0
0.05
0.1
0.15
0.2
0.25
0.3
1.695
1.962
2.199
2.408
2.599
9.63
9.86
9.84
9.49
9.6
Attal 5
Run 3:
Time (s)
Elapsed
Time (s)
Run 3:
Run 3:
Run 3:
Run 4:
Distance Velocity Acceleration Time (s)
(m)
(m/s)
(m/s²)
0.623502
0.647021
0.668384
0.688084
0.706384
0.723684
0.740008
0
0.023519
0.044882
0.064582
0.082882
0.100182
0.116506
0
0.05
0.1
0.15
0.2
0.25
0.3
2.238
2.443
2.639
2.813
2.979
9.56
9.63
10.21
8.88
10.28
Table 3 -Part 2, Raw Data:
Run 1: Distance from
Run 2: Distance from
camera (cm)
camera (cm)
10
20
0.731216
0.751283
0.7699
0.787484
0.804083
0.8199
0.835023
Elapsed
Time(s)
0
0.020067
0.038684
0.056268
0.072867
0.088684
0.103807
Run 4:
Run 4:
Run 4:
Distance Velocity Acceleration
(m)
(m/s)
(m/s²)
0
0.05
0.1
0.15
0.2
0.25
0.3
Run 3: Distance from
camera (cm)
30
2.592
2.767
2.93
3.089
3.235
10.03
8.73
9.86
9.2
9.37
Run 4: Distance from
camera (cm)
40
Data Analysis
PART 1: REPEATABILITY
Figure 5 – Velocity vs. Time Graph of Photogate Freefall
VELOCITY VS. TIME GRAPH OF PHOTOGATE
FREEFALL
Trial 1
Trial 2
Trial 3
Linear (Trial 1)
Linear (Trial 2)
Linear (Trial 3)
Trial 1
y = 9.7197x + 0.6328
R² = 1
3
VELOCITY (M/S)
2.5
Trial 2
y = 9.6724x + 0.7251
R² = 1
2
1.5
1
Trial 3
y = 9.7365x + 0.7174
R² = 1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
TIME (S)
The slope of each line is extremely close to the value of gravity (9.8 m/s 2). For example, Trial 1 has a
slope of 9.7197, and Trial 2 has a slope of 9.6724. However, the mean slope is 9.7095 (work shown in
error analysis) with a maximum slope from Trial 3 (9.7365) and minimum from Trial 2 (9.6724). Due to
gravity acting as a constant form of acceleration, the velocity also increases as the photogate descends.
However, the constant only show the initial velocity of each trial once it first passes through the
photogate. The initial velocities are slightly different from each other simply because of the difficulty of
dropping the picket fence from the same exact spot every time. The greater the height from the starting
point for the free fall is, the greater its initial velocity will be since it has more distance to accelerate and
increase its speed. This indicates that Trial 2 started from the highest point above the photogate since it
has the greatest initial velocity of 0.7251 m/s. Trial 1 differed slightly with a constant of 0.6328.
Attal 6
.
PART 2 : VARYING HEIGHTS
.
Figure 6-Distance vs. Time Graph of Photogate Free Fall
Distance vs. Time Graph of Photogate Free Fall
0.35
Trial 1
Trial 2
Trial 3
0.3
Trial 4
Poly. (Trial 1)
Poly. (Trial 2)
0.25
Poly. (Trial 3)
Poly. (Trial 4)
Distance (m)
0.2
Trial 1
y = 4.8933x2 + 0.581x - 2E-05
R² = 1
0.15
0.1
Trial 2
y = 4.8625x2 + 1.3803x - 5E-06
R² = 1
0.05
0
0
-0.05
0.05
0.1
0.15
Time (s)
0.2
Trial 3
y = 4.8426x2 + 2.0106x + 8E-06
R² = 1
0.25
Trial 4
y = 4.7015x2 + 2.4023x - 3E-05
R² = 1
Attal 7
Figure 7: Velocity vs. Time Graph of Photogate Free Fall
Velocity vs. Time Graph of Photogate Free Fall
Trial 1
4
Trial 2
Trial 3
3.5
Trial 4
Linear (Trial 1)
3
Linear (Trial 2)
Linear (Trial 3)
Velocity (m/s)
2.5
Linear (Trial 4)
2
Trial 1
y = 9.7643x + 0.5838
R² = 1
1.5
Trial 2
y = 9.7243x + 1.3803
R² = 1
1
Trial 3
y = 9.6812x + 2.0105
R² = 1
0.5
0
0
0.05
0.1
0.15
Time (s)
0.2
0.25
Trial 4
y = 9.3806x + 2.4037
R² = 1
For the first graph (distance vs. time), one trend is that each successive trial appears to become
steeper. This situation is due to the fact that the velocity of the photogate was higher when there was more
distance for the photogate to travel through as well as more distance for gravity to accelerate it. The
velocity is shown from the linear coefficient so Trial 1 has a velocity of 0.581 m/s while Trial 2 has a
velocity of 1.3803m/s, showing an increase in velocity. In addition, another feature is that the quadratic
coefficient for the trials is half of the standard value of gravity since Trial 1 has a value of 4.8933 and
Trial 2 a value of 4.8625. However, all the y-intercept values are very nearly zero because all trials of the
photogate began at the same point (the camera). Both Trial 2 and Trial 3 have constants infinitely close to
0, 5•10-6 and 8•10-6, respectively. In addition, the linear coefficient increases with every run due to the
increased velocity for each trial. Just as how each curve for the distance-time graph was steeper, each
linear coefficient was higher due to a greater velocity from a higher distance. This trend is also continued
in the velocity-time graph; however, the velocity is shown as a constant. For example, for Trial 1, the
constant is 0.5838 and for Trial 2, 1.3803, values that are very close to the linear coefficients in the
distance-time graph. However, for the velocity-time graph, the acceleration (linear coefficient) is shown
to be double the value of the quadratic coefficient, showing a closer precision to the actual value of
gravity. For the velocity-time graph, not only does each run starts at a higher velocity for the y-axis, but
due to this higher velocity, each run appears to “start” at an earlier time on the x- axis because the time
between passing the first two parts of the photogate (which is the same distance) is less for each trial due
to the higher velocity in successive trials.
Attal 8
Error Analysis
PART 1: REPEATABILITY
Average acceleration (g) = Sum of all linear coefficients of velocity-time graph / 3
= (9.7197 + 9.6724 + 9.7365) / 3
= 9.7095 m/s2
Precision = |max g – min g|
——————
2
= | 9.7365-9.6724|
——————
2
= 0.03
Precision of g = 9.71 ± 0.03 m/s2
Accuracy = |experimental value- accepted value|
—————————————— *100
Accepted value
= |9.7095 – 9.8|
—————— *100
9.8
= 0.92% error
Overall, the results of this part of the experiment yield very precise measurements. The precision
is only 0.03, meaning that each of the trials only vary from each other by a slight amount,
making all of the slopes exceedingly close (as can be seen on the graph). The average value of
the results (9.7095) also demonstrates great accuracy to the actual value of gravity (9.8 m/s2),
with a minute percent error of just 0.92%.
PART 2: VARYING HEIGHTS
Average acceleration (g) = Sum of all linear coefficients of velocity-time graph/4
= (9.7643+9.7243+9.6812+9.3806)/4
= 9.6376 m/ s2
Precision = |max g – min g|
——————
2
——————— * 100
Average g
Attal 9
= | 9.7643-9.3806|
——————
2
—————— * 100
9.6376
= 2% (0.2)
Precision: 9.6376 ± 0.2 m/ s2
Accuracy= |accepted value- experimental value(average)|
——————
*100
accepted value
= |9.6376 – 9.8|
—————— *100
9.8
= 1.7%
Accuracy of max experimental g value= |accepted value- experimental value(max)|
——————
*100
accepted value
=
|9.8– 9.7643|
—————— *100
9.8
= 0.36%
Accuracy of min experimental g value= |accepted value- experimental value(min)|
——————
*100
accepted value
=
|9.8– 9.3806|
—————— *100
9.8
= 4.3%
In the data for the velocity-time graph, each successive trial has an acceleration (linear
coefficient) that deviates more and more from the accepted value of gravity (9.8m/ s2). For example,
while Trial 1 has an acceleration of 9.7643 m/ s2, Trial 4 has an acceleration of 9.3806 m/ s2.
However, the velocity (constant) in each successive trial increases. As for the precision of the lab,
each g value was only 2% at the most from the average g value of 9.6376 m/ s2. These results are
satisfactory and indicate a strong trend within the results. In addition, the accuracy of the average
value was only 1.7% from the accepted value of gravity. However, one observation to note is that
while Trial 1 with an acceleration of 9.7643 m/ s2 and accuracy of 0.36%, Trial 4 had an acceleration
of 9.3806 m/ s2 and accuracy of 4.3%, a noticeable jump in accuracy.
Attal 10
Conclusion
The purpose of this experiment was to find the acceleration of free fall as well as evaluate the
relationships of velocity, distance, and acceleration. By using the results of the photogate experiment,
the experimental value of acceleration is 9.6376 m/ s2 by using four trials of different distance and
9.7095 m/ s2 by using three trials of similar height. While it has been proven that the actual value of
gravity is 9.8 m/ s2, air resistance had to be taken into account, explaining the 1.7% deviation for
varying heights; however, that is not to say that the results were not close, especially for the
repeatability portion of the experiment with a percent error of 0.92%. Not only were the results fairly
accurate, but the values of the experiment showed great precision and repeatability, especially in Part
1 of the experiment with a precision of only ±0.03 m/ s2. However, there are multiple notable errors in
this experiment. For example, for future experiments, it would be better to use a machine that drops
the photogate at an exact height so that the precision for the acceleration of the trials will be less. In
addition, using the human eye to judge 10 or 40 cm from a ruler is not advisable for precise results.
Ultimately, the experiment still supported the hypothesis of the experiment and proved that not only
is the acceleration of an object in free fall close to the value 9.8 m/ s2 , but also that the velocity
increases from a greater time in acceleration.
.