A lab activity designed for introductory physics students to compare measured and calculated ranges for a projectile launched from a reference height is presented.
A lab activity designed for students to compare measured and calculated ranges for a projectile launched from a reference height is presented here. Students used statistical and error propagation techniques to analytically determine the error bounds associated with measured and calculated projectile ranges as well as t-statistics to determine how well the measured and calculated ranges agreed. For all launch angles used in this work, 90% of students found that there was no statistically significant difference
Every physical measurement has an associated error. Including error in reported physical results establishes a basis for deciding whether a scientific hypothesis should be accepted or rejected. As such, reporting errors establishes a level of confidence associated with the measured value, reflects the quality of the experiment, and allows for comparison with theoretical values. Apart from reporting the errors, understanding error propagation is important as it adds validation to reported experimental results (Chhetri, 2013; Baird, 1995; Labs for College Physics, n.d.; Monteiro et al., 2021). The error propagation technique is a skill that the American Association of Physics Teachers (AAPT) recommends that all Physics students develop (AAPT, 2014). Experimental data are routinely used to make conclusions in physics, astronomy, chemistry, life sciences, engineering and many other fields of study. Error propagation techniques are particularly helpful to students or experimentalists in these fields to apply best practices to make estimates for experimental measures and report inaccuracies within practical reasons. (Lippmann, 2003 & Berendsen, 2011). However, error propagation techniques are rarely emphasized in introductory physics classes, as such, few undergraduate lab activities exist that focus on these techniques (Taylor, 1985; Allen, 2021). In fact, error propagation may only be included in junior/senior level physics classes and are largely omitted or greatly simplified in introductory classes (Faux & Godolphin, 2019; Purcell, 1974).
This paper presents a lab activity designed to introduce students to error propagation methods as applied to the range measurements of a projectile. Students measured ranges of projectile launched at various angles from a reference height and compared their measured results to calculated ranges by employing statistical and error propagation techniques. Students then used t-statistics to establish the level of confidence in their measurements. This lab activity was implemented in an introductory physics course at Georgia Gwinnett College (GGC). Georgia Gwinnett College is an access, four-year, minority-serving, Hispanic-serving, liberal arts, public institution committed to student success. The college emphasizes an integrated educational experience for students and encourages new teaching pedagogy as well as the innovative use of technology. Physics class sizes at GGC are limited to a maximum of 24 students and taught in a studio-style setting in 2 hours and 45-minute sections.
The experimental data collection for this activity took 45 minutes and the remainder of the class time was spent on data analysis and completing the lab report. The authors intend that upon completion of this activity, students understand how to: 1) calculate averages and standard errors from experimental data, 2) apply error propagation techniques, 3) report errors in measurements, and 4) use t-tests to compare the measured and calculated results. Typically, GGC students enrolled in introductory calculus-based physics classes are pre-engineering majors and have only completed Calculus I and have not necessarily taken a statistics course.
The purpose of this lab activity is to introduce students in introductory physics courses how to apply error propagation techniques in establishing error bounds in experimental measurements.
Typically, most students enrolling in introductory calculus-based physics classes have only completed calculus I and have not necessarily taken a statistics course. The authors intend that upon completion of this activity, students would be able to understand how to 1) find averages and standard errors from experimental data, 2) report errors in measurements, 3) apply error propagation techniques, and 4) use t-tests to compare the measured and calculated results.
The materials required for this lab include a PASCO projectile launcher fitted with a protractor (ME-6800), small metal PASCO launch balls (ME-9859), meter stick, carbon paper, and safety glasses.
Prior to the start of the experiment, the concepts of projectile motion were reviewed. This review with students allowed for further questions, discussions, and improved understanding of projectile motion. The overviews shared with students are presented in the following two sections (Overview 1 and Overview 2).
Figure 1 shows the variables associated with a projectile launched horizontally from a known height with velocity (
For such a horizontal launch,
The time of flight is obtained from equation (2) as:
The expression for
Figure 2 shows the trajectory of a projectile launched at an angle (
The kinematics equations in the y-direction can be written:
Where
Additionally, for conciseness, the basic statistical formulas needed to calculate the average and standard errors of measurements, t-statistics for comparing two values, and error propagation rules are summarized in Appendix 1. Appendix 2 is provided to show the derivation of the equations used in calculating the errors associated with launch velocity, time of flight, and calculated range (
The lab activity is comprised of four parts:
Part I: Determine the initial launch velocity,
Part II: Measure projectile range (
Part III: Calculate ranges (
Part IV: Compare measured ranges ( Δx_{meas}) to the calculated ranges (
A projectile was launched from a table onto the floor as shown in the experimental setup in Figure 1. Using a meter-stick with a 0.05 cm accuracy, the range (
Following the experimental set-up shown in Figure 2, a projectile was launched at three different angles and
This part dealt with calculated ranges,
Determined the initial x- and y-components of the launch velocities (
Found the time of flight (
Calculated,
and recorded the calculated range in the form:
Table 3 lists the calculated values of
The data presented in this section represents the measured projectile ranges launched horizontally (0.0^{o} ) and ranges for projectile launched at different angles (35.0^{o}, 45.0^{o}, and 55.0^{o}). Also, the calculated projectile ranges and initial launch velocities at the different launch angles are presented. Furthermore, the calculated projectile time of flight, with associated error is presented. Finally, an analysis of the measured range is compared to the calculated range using t-statistics. The provided data are complete and match the descriptions in the contribution. Readers who are interested in obtaining a copy of the lab materials provided to students should contact Joseph Ametepe ([email protected]).
TABLE 1: Measured ranges for horizontal launch
Trial | $\overline{\mathrm{\Delta}x} \pm \delta(\overline{\mathrm{\Delta}x})(m)$ | $\overline{\mathrm{\Delta}y} \pm \delta(\overline{\mathrm{\Delta}y})(m)$ |
---|---|---|
Average | $(1.\ 342 \pm 0.002)$ | $1.0980\ \pm 0.0005$ |
The above measured values of
TABLE 2: Measured ranges for projectile launched at different angles (precision of compass
Angle | Measured Values | |
---|---|---|
35.0^{o} ± 0.5^{o} | $1.544\ \pm \ 0.008$ | $1.0980\ \pm 0.0005$ |
45.0^{o} ± 0.5^{o} | $1.440\ \pm \ 0.009$ | $1.0980\ \pm 0.0005$ |
55.0^{o} ± 0.5^{o} | $1.23 \pm 0.03$ | $1.0980\ \pm 0.0005$ |
It should be noted that the error in the 55-degree measurement is significantly higher than for the other two angles. This is at least partially due to the fact that the work done by the spring has been neglected in our calculations and here it is non-negligible (Schnick, 1994).
Table 3 shows a sample of student data listing the calculated values of
TABLE 3: Initial launch velocities (
Angle | $v_{ox} = v_{o} \cos\theta$ | $v_{oy} = v_{o} \sin\theta$ | $(v_{oy} \pm \delta\left( v_{oy} \right))(\frac{m}{s})$ | |
---|---|---|---|---|
35.0^{o} ± 0.5^{o} | 2.3223 | 1.6261 | $2.32\ \pm 0.02$ | $1.63\ \pm 0.02$ |
45.0^{o} ± 0.5^{o} | 2.0046 | 2.0046 | $2.00\ \pm 0.02$ | $2.00\ \pm 0.02$ |
55.0^{o} ± 0.5^{o} | 1.6261 | 2.3221 | $1.63\ \pm 0.02$ | $2.32\ \pm 0.01$ |
Table 4 lists the time of flight
TABLE 4: Calculated
Launch angle ( | $\delta t_{cal}$ | $t_{cal} \pm \delta t_{cal}$ | |
---|---|---|---|
0.6675 | 0.0022 | $0.668 \pm 0.002$ | |
0.7202 | 0.0020 | $0.720 \pm 0.002$ | |
0.7663 | 0.0016 | $0.766 \pm 0.002$ |
Table 5 lists the calculated range values (
TABLE 5: Calculated range,
Angle | $\left\lbrack v_{ox}\ \pm \delta\left( v_{ox} \right) \right\rbrack(\frac{m}{s})$ | $\left\lbrack t_{cal} + \delta t_{cal} \right\rbrack(s)$ | ${\mathrm{\Delta}x}_{cal}(m)$ | $\delta(\mathrm{\Delta}x_{cal})$ | ${\mathrm{\Delta}x}_{cal} \pm \delta(\mathrm{\Delta}x_{cal})$ |
---|---|---|---|---|---|
35.0^{o} | $2.32\ \pm 0.01$ | $0.668 \pm 0.002$ | 1.5502 | 0.0110 | $1.55 \pm 0.01$ |
45.0^{o} | $2.00\ \pm 0.02$ | $0.720 \pm 0.002$ | 1.4438 | 0.0134 | $1.44 \pm 0.01$ |
55.0^{o} | $1.63\ \pm 0.02$ | $0.766 \pm 0.002$ | 1.2461 | 0.0159 | $1.25 \pm 0.02$ |
TABLE 6: Measured and calculated ranges, and
Angles | ${\mathrm{\Delta}x}_{cal} \pm \delta(\mathrm{\Delta}x_{cal})$ | $p = \frac{{\mathrm{\Delta}x}_{m - c}}{\delta\left( {\mathrm{\Delta}x}_{m - c} \right)}$ | ||
---|---|---|---|---|
35^{o} | $1.545 \pm 0.009$ | $1.55 \pm 0.01$ | $0.0055$ | $0.49191$ |
45^{o} | $1.440 \pm 0.009$ | $1.44 \pm 0.01$ | $0.0038$ | $0.280672$ |
55^{o} | $1.23 \pm 0.03$ | $1.25 \pm 0.02$ | $0.0112$ | $0.701229$ |
in this part, students compared
Table 6 shows measured and calculated ranges from Tables 3 and 4, and
Data in Table 2 clearly show that the lunch angle of
Students were encouraged to explore the meaning of their t-statistics values, especially the ratio of
It should be noted that students can complete many of the calculations presented in this work by hand, however, they can also use other programs such as Mathcad or Excel (Gardenier et al., 2011; Donato & Metz, 1988; de Levie, 2000).
In Part I of this activity, students investigated a projectile fired horizontally from a reference height. With their recorded range and launch height, they were able to calculate the initial launch velocity,
Student data for launch angles 35^{o}, 45^{o}, and 55^{o} is given in Table 6.
For all launch angles, 90% of students reported that there was no significant difference between the measured and calculated ranges. This was valid within the error bounds established by the error propagation techniques employed.
Through this activity, our students learned transferable skills of how to calculate averages and standard errors from repeated measurements and how to use t-statistics to compare measured and calculated values. Furthermore, students learned the condition of maximum projectile range occurring at 45 degrees applies only when the projectile is fired from ground level.
Completing this activity presented students with transferable skills and learning opportunities associated with experimental measurements, applying error propagation, theoretical calculations of ranges for projectile launched from a reference height, and applying t-statistics to compare two values.
Specifically, students understood (from the set up in Figure 1) that the initial launch velocity can be determined from range and launch height measurements. Furthermore, students were able to apply error propagation techniques to determine error-associated boundaries for their projectile range measurements and calculations. The error propagation exercise portion was an important component of this activity as students do not traditionally cover error propagation in many introductory lab activities (Taylor, 1985; Allen, 2021; Faux & Godolphin, 2019; Purcell, 1974).
Another learning opportunity was associated with the use of t-statistics to compare the measured and calculated ranges. For all launch angles (35^{o}, 45^{o}, and 55^{o}) used in this work, 90% of students found that there was no statistically significant difference
The results of the small group of students who had statistically significant differences between calculated and measured range values were investigated. A common error that occurs when students perform projectile motion labs is the failure to ensure that the exit of the launcher barrel is consistently set as the origin of measurements. This critical issue led to values outside the error bounds. If the barrel extends beyond the origin of measurement, errors due to a subtle conservation of energy considerations that needs to be compensated for are introduced. Compensating for these errors will be the focus of a forthcoming paper.
This Appendix summarizes simple statistics and error propagation methods.
Presented here is a review of finding the mean, standard deviation and associated errors. Suppose we measure a quantity N times (N ≥ 10), then
The mean (
The standard deviation σ of the sample measurements can be written as:
The “error” in the measurement is the standard error (
The mean value of measurement with error is reported as:
The t-statistics to determine whether the difference between two independent values
where,
and,
where
TABLE 7: Meaning of p values
$\mathbf{p =}\overline{\mathbf{\mathrm{\Delta}x}}\mathbf{/}\mathbf{\sigma}_{\mathbf{\mathrm{\Delta}x}}$ | Conclusion |
---|---|
$\mathbf{p < 1}$ | there’s no significant difference; |
$\mathbf{1 < p < 2}$ | the determination is inconclusive |
$\mathbf{2 < p < 3}$ | the measurements are different with better than 95% confidence |
$\mathbf{3 < p < 4}$ | the measurements are different with better than 99.5% confidence |
$\mathbf{4 < p < 5}$ | the measurements are different with better than 99.994% confidence |
When dealing with error propagation, there are specific rules (Harvey, 2009; Allen, 2021) to be followed that are summarized in A1.8 – A1.14. In this activity, we assumed that the variables are uncorrelated and, therefore, the error is evaluated using simple sums of partial derivatives (Taylor, 1997). If
Addition
Multiplication
Division
Multiplying by a constant,
Single-valued functions: For a single-valued function R(X), the associated error
Multi-valued Functions: For a multi-valued function R(X, Y, Z, …), in which X, Y, Z, … are uncorrelated, then:
Polynomial Functions (R based on polynomial function of one variable X)
The initial launch velocity and associated error is derived by starting with equation (4).
so,
where
Using equation (A1.9),
with
Therefore,
Now, we show how
Similarly, applying equation (A2.2) to
Next, we derive the equations necessary to determine the time of flight and its associated error. Using equation (6), the time of flight for the projectile launched at an angle (
To find
where
Applying equation A1.9 to
Substituting equations (A2.7) and (A2.11) in equation (A2.10),
Finally, the derivation of the error in ∆x_{cal} is derived starting from
The authors want to acknowledge and thank the School of Science and Technology at Georgia Gwinnett College for providing funds for the purchase of materials to support this activity.
Allen, T. J. (2021). Experimental Determination of Projectile Range, Retrieved April 26, 2020 from people.hws.edu › tjallen › Physics150 › labreportexample
American Association of Physics Teachers (AAPT). 2014. AAPT recommendations for the undergraduate physics laboratory curriculum. https://www.aapt.org/Resources/upload/LabGuidlinesDocument_EBendorsed_nov10.pdf
Baird, D. C. (1995). Experimentation: An Introduction to Measurement Theory and Experiment Design, (3^{rd} ed.) Prentice Hall. https://doi.org/10.1063/1.2808025
Berendsen, H. J. C. (2011). A Student’s Guide to Data and Error Analysis. Cambridge University Press, New York. www.cambridge.org/9780521119405
Chhetri, K. B. (2013). Computation of errors and their analysis in physics experiments. The Himalayan Phys., 3, 78-86. https://doi.org/10.3126/hj.v3i0.7312
de Levie, R. (2000). Spreadsheet Calculation of the Propagation of Experimental Imprecision. J. Chem. Ed. 77 (4), 534. https://doi.org/10.1021/ed077p534
Donato Jr., H. and Metz, C. (1988). A Direct Method for the Propagation of Error Using a Personal Computer Spreadsheet Program. J. Chem. Ed. 65 (10), 867. https://doi.org/10.1021/ed065p867
Faux, D. A. & Godolphin, J. (2019). Manual timing in physics experiments: Error and uncertainty. Am. J Phys. 87, 110-115. https://doi.org/10.1119/1.5085437
Gardenier, G. H., Gui, F., & Demas, J. (2011). Error Propagation Made Easy – Or at Least Easier. J. Chem. Ed. 88 (7), 916-920. https://doi.org/10.1021/ed1004307
Harvey, D. (2009). Analytical Chemistry 2.0. Retrieved May 23, 2023 from https://chem.libretexts.org/Under_Construction/Purgatory/Book%3A_Analytical_Chemistry_2.0_(Harvey)
Labs for College Physics: Mechanics 2nd Edition (n.d.). Retrieved April 26,2020 from https://www.webassign.net/question_assets/unccolphysmechl1/measurements/manual.html.
Lippmann, R. F. (2003). Students' understanding of measurement and uncertainty in the physics laboratory: Social construction, underlying concepts, and quantitative analysis (Order No. 3112498). Available from ProQuest Dissertations & Theses A&I. (305321079). Retrieved from https://www.proquest.com/dissertations-theses/students-understanding-measurement-uncertainty/docview/305321079/se-2
Monteiro, M., Stari, C., Cabeza, C. & Marti, A. C. (2021). Using mobile-device sensors to teach students error analysis. American Journal of Physics 89 (5), 477-481. https://doi.org/10.1119/10.0002906
Purcell, J. E. (1974). Particle Diffraction by a Single Slit as Error Propagation, Am. J of Phys. 42, 1112-1116 . https://doi.org/10.1119/1.1987949
Schnick, J. (1994). Projectile motion details. The Physics Teacher 32, 266 – 269. https://doi.org/10.1119/1.2343992
Taylor, J. (1985). Simple examples of correlations in error propagation. Am. J Phys. 53, 663-667. https://doi.org/10.1119/1.14281
Taylor, J. (1997). An Introduction to Error Analysis, (2^{nd} ed). University Science Books.