General guidelines for lab reports
1) Each person will write their own report. Include the names of everyone who
contributed.
2) Reports will be typewritten and include tables and graphs, as appropriate, to
demonstrate the work and support the conclusions.
3) There are no particular font, margins, or pages requirements.
4) A complete report should include the following sections:
• Abstract States the main goal and the main results.
• Introduction Describes why the experiment is being performed.
• Methods & Data Describes the experimental setup in both words and with
appropriate graphics. This section may also include formulae or derivations
needed to demonstrate the objectives of the experiment, and tables of data
or derived parameters.
• Analysis Interprets the data. This section may also describe the precision of
the results achieved and the main sources of error or uncertainty. This
section should include graphs or figures that help interpret the data.
Equations or derivations using basic data to compute other parameters may
also be included here.
• Results & Conclusions Summarizes the results and describes what worked
well or what could have been changed to achieve better results.
• Appendix Includes work performed but not essential to the main report,
e.g., computer code for making plots or doing computations.
5) Include a photo of pertinent aspects of your setup or equipment. Drawings are
also often helpful, as they can be labeled to show sizes, distances, etc.
6) The text should follow standard English grammar, punctuation, and sentence
structure.
7) You should share data among the group but make your own tables and plots!
Your report should be unique and reflect your own writing and thinking.
Uncertainty vs Error
I'm using the term "uncertainty" in the standard way: to reflect our confidence in a
quantity. I'm using the term "error" to only reflect the percentage difference from
expectations. Uncertainties should be provided for every number including the final
result; errors should be additionally provided for the final result.
Below is a note I sent to my students in Spring 2018 for the Lab 2 report.
Example #1 for uncertainty
For something directly measured like the mass of a cube, the "uncertainty" is your best
estimate about the accuracy of the scale, maybe ± 1 gram.
Example #2 for uncertainty
For something that relies on both measurements and an equation, like the average
volume of the gardyloo, the uncertainty can be computed via
ε(⟨V⟩) = σ/sqrt(3),
in other words, the standard deviation of your individual volumes divided by the square
root of the number of trials you carried out. One can also estimate the uncertainty via
propagation of errors, like we did for Lab #1, but typically we'll use the statistical
approach.
Example #1 for error
I expect to see an error provided for a lab's final result, like the molar mass. In this
example, the error can be computed via the percent difference from expectations:
error(M) = 100*|M- Mknown|/Mknown
For Lab #2
We didn't have time to carry out three independent trials for M, and because the
propagation of errors is messy, I don't expect an uncertainty on M. And since we can't
look up a reference value for Vgardyloo, I don't expect an error for ⟨Vgardyloo⟩. In short, I
only expect an error for M and an uncertainty for ⟨Vgardyloo⟩.
Lab Report Grading Rubric
04% Signatures on Experimental Plan & Final Data
08% Abstract: States the main goal and the main results.
18% Introduction: Describes why the experiment is being performed.
20% Methods & Data: Describes the experimental setup and execution. Includes tables of
measurements and graphics/figures that demonstrate the methods and interpret the data.
20% Analysis: Derives and interprets the results using equations and graphs that demonstrate the
objectives of the experiment. Describes the precision of the results achieved and the main sources of
uncertainty.
20% Results and Conclusions: Summarizes the results and their uncertainties, and describes what
worked well or what could have been changed to achieve better results.
10% The report is neat and legible and shows original thought and understanding. The work is not
copied from a friend or a solutions manual.
Experiment 999
Measuring an Acceleration
Ima Genius
in collaboration with
B.A. Helper, M.Y. Partner, A. Dude
Abstract
We measured the acceleration of a HotWheels car down an incline of
constant slope. Observers recorded the distance traveled by the car at
time intervals of 0.5 seconds over the four seconds required for the car to
reach the bottom of the ramp. The position-time data were used to
compute the average velocity in each of eight 0.5-second time intervals.
We then used the change in velocity over each 0.5-second time interval to
compute the acceleration. Our average acceleration over the eight
intervals was 0.469 m/s2 with a standard deviation of 0.099 m/s2 and thus
an error of 0.035 m/s2.
Introduction
Acceleration is a change in the velocity of an object. Generally, an average acceleration may be
expressed as a change in velocity, Dv, over some time interval, Dt.
D v
aavg=D t
Or, in the limit that Dt goes to zero,
a d vavg= d t
Measuring an acceleration, therefore, requires measuring a change in the position of an object and
timing the duration required for each movement. In this experiment, we chose to measure the
acceleration of a race car (really a HotWheels car) down an inclined road (really an orange track). Our
goal was to measure the acceleration of the car to a precision of at least 0.01 meters per second squared
by sing multiple measurements of the car's position over several seconds.
Methods
We set up the race track on an inclined slope made by two metal tracks supported by bricks. Each
metal track was two meters long, so that they form a solid surface four meters long when placed end-to
end. Upon this solid surface we placed several sections of HotWheels track connected together. The
track drops roughly 18 inches over four-meter length. Although we could measure the incline angle to
be roughly 15 degrees, it was not necessary for this experiment. Figure 1 below shows the
experimental setup.
The metals ramps are
4 meters
Support Metal ramp topped
bricks by plastic race
tracks
Figure 1: Schematic of the experimental setup
marked with distance in cm along one edge, making it easy to measure the position of the car at any
point along the ramp.
We released the car from rest at the top of the ramp with the rear end of the car at the zero mark. At
each half second interval (e.g., 0 s, 0.5 s, 1.0 s, etc...) we measured the position of the rear edge of the
car. Three or four people measured the position of the car at each time interval. We found that was
possible to measure the position to an accuracy of at least 1 cm when the car was moving slowly, and
each person estimated the position to 1.10 of a cm or 1 mm. However, once the car was moving more
quickly, it became harder to measure the position with similar accuracy. We estimate that the positions
are accurate to no more than 1 cm. Because several people took data at each time, we record all of
their measurements, and we computed an average position at each time in order to help reduce random
measurement errors. Table 1 below shows the measurement from each person and the average position
of the car at each time.
T able 1. Posit ion, Velocity, Accelerat ion data
Measured Data Calculated parameters
Distance Distance DistanceDistanceDistance Avg. Velocity Avg. Acceleration
T ime (m) (m) (m) (m) (m) (m/s) (m/s2)
(s) Person 1 Person 2 Person3 Person4 Average
0.00 0.000 0.000 0.000 0.000 0.000 0
0.50 0.065 0.056 0.058 0.069 0.062 0.124 0.248
1.00 0.251 0.262 0.240 0.248 0.250 0.3765 0.505
1.50 0.562 0.540 0.578 0.569 0.562 0.624 0.495
2.00 1.051 1.011 0.980 0.991 1.008 0.892 0.536
2.50 1.569 1.530 1.630 1.590 1.580 1.143 0.502
3.00 2.256 2.267 2.244 2.239 2.252 1.3435 0.401
3.50 3.067 3.040 3.079 3.050 3.059 1.615 0.543
4.00 4.010 4.060 3.950 3.967 3.997 1.8755 0.521
Average 0.469
StdDev 0.099
Analysis
Figure 2 shows a plot of the position of the car
versus time. The plot shows that the car moved
farther in any given time interval as the car moved
along the ramp. The shape of the curve is roughly
that of a parabola.
We computed the velocity of the car during each
0.5 s time interval by taking the distance traveled,
Dx, divided by the time interval, Dt.
(x −x )
v (m /s)=D x= 2 1D t ( t2−t 1)
For example, after the first 0.5-second time Figure 2: Position of the car in meters versus time
interval, the velocity is using data from Table 1.
(0.062−0.000)
D x= =0.124 m /s
D t (0.5−0.0)
The 6th column of Table 1 lists the velocities computed in this manner from each time intervals. Figure
3 below shows a plot of velocity versus time using the data from Table 1. The trend is a approximately
a straight line with a positive slope, indicating increasing velocity. A straight line is consistent with a
constant acceleration given by the slope of the line. The average velocity of the entire journey is
3.997 meters / 4.0 seconds = 1.00 m/s.
However, the instantaneous velocity is much
smaller during the first portion of the
experiment and much larger during the latter
portion.
Finally, we used the velocity data to compute
the acceleration during each time interval.
The average acceleration was computed from
a D v
(v2−v1)
avg= =D t ( t2−t 1)
For the first time interval this yields
(v2−v1) (0.124−0.000)= =0.248 m /s2 Figure 3: Velocity of the car versus time.
(t2−t1) (0.5−0.0)
The computed accelerations appear in Column 7 of Table 1. The values are relatively constant and fall
near 0.5 m/s2. At the bottom of Table 1 we compute the average acceleration over the eight time
intervals to be 0.469 m/s2 and the standard deviation of this set of data to be 0.099 m/s2.
Results and Conclusions
We measure the car's acceleration to be a=0.469±0.099 m/s2.. We use this value to compute the
theoretical distance versus time curve using
x1=x0+ v t+
1
0 a t
2
2 ,
where v0=0 is the car's initial velocity and x0=0 is the car's initial position. Figure 4 shows the position-
time plot of the data (asterisks) and the theoretical position-time curve (solid line). There is good
agreement between the data and the curve, suggesting that the stated acceleration is a good
representation of the car's motion overall. However, the curve falls increasingly below the data at later
times, suggesting that we have underestimated the acceleration. We notice that our first computed
acceleration in Table 1 is about half of the other values. If we reject this one measurement and
compute an average of the remaining data, we find an average acceleration of 0.501, which would
provide a better fit to the data. We don't have a good explanation for why our first acceleration value is
low compared to the others, but it may have to do with difficulties in timing the very first measurement
at t=0.5 s. We conclude that we have measured the acceleration reasonably well.
In order to make a better measurement we might use a longer track so that we would have even more
data points and a longer time over which to conduct the experiment. We could also have more people
time the car at various points along the track so that, with more measurements, our data would be more
accurate over each time interval. We could also try using a steeper track. We suspect that perhaps
friction in the car's wheels would prevent the car from accelerating as quickly as it might otherwise if
the track were steeper.
Figure 4: The position of the car versus time, showing the
data (asterisks) and the theoretical curve computed using the
average acceleration (solid line).
Appendix
Below we show the Matlab code used to make the plots in Figures 2,3,4.
% make distance time plot in Figure 2
t=[0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0]
x=[0.000,0.062,0.250,0.562,1.008,1.580,2.252,3.059,3.997]
plot(t,x,'bs')
xlabel('Time (s)')
ylabel('Distance (m)')
% make velocity-time plot for Figure 3
t=[0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0]
v=[0.0,0.124,0.376,0.624,0.892,1.143,1.343,1.615,1.876]
plot(t,v,'rs')
xlabel('Time (s)')
ylabel('Velocity (m/s)')
% make distance-time plot for Figure 4 with theoretical curve overplotted
t=[0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0]
x=[0.000,0.062,0.250,0.562,1.008,1.580,2.252,3.059,3.997]
plot(t,x,'rs')
hold on
a=0.469
plot(t,0.5*a*t.^2)
xlabel('Time (s)')
ylabel('Distance (m)')