Error propagation in projectile motion lab

Abstract

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 $(p < 1)$ between the measured and calculated ranges.

Introduction

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.

Purpose

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.

Learning objectives

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.

Materials and Equipment

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.

Procedure

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).

Overview 1

Figure 1 shows the variables associated with a projectile launched horizontally from a known height with velocity ($v_{o})$.

For such a horizontal launch, $\mathrm{\Delta}x$ is the horizontal distance (range) covered by the projectile, $\mathrm{\Delta}y$ the known launch height, $v_{ox}$ and $v_{oy}$ are the x- and y-components of initial launch velocity respectively, and $t$ is the time of flight. When kinematic equations are applied to projectile motion with constant downward acceleration (g= 9.81 m/s²), the expressions for $\mathrm{\Delta}x$ and $\mathrm{\Delta}y$ become:

The time of flight is obtained from equation (2) as:

The expression for $v_{o}$ can be obtained by combining equations (1) and (3):

Overview 2

Figure 2 shows the trajectory of a projectile launched at an angle ($\mathrm{\theta=0}$ ) with respect to the horizontal, from a reference height ($\mathrm{\Delta}y=-h$ ). In this experiment, air resistance is considered to be negligible and g=9.80 m/s².

The kinematics equations in the y-direction can be written:

 $v_{y} = v_{0y} - gt$ (5)

$\mathrm{\Delta}y = h + v_{0y}t - \frac{1}{2}gt^{2}$ (6)

$v_{y}^2 = v_{0y}^2 - 2g(y-h)$ (7)

Where  ${v_{0y} = v_{0}\sin(\theta)=0}$ and $v_{y}$ is the y-components of the projectile’s velocity at any point along the trajectory.

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 ($\delta v_{o},\ \ \delta t_{o},\ \ and\ \delta(\mathrm{\Delta}x_{cal})$), respectively.v

The lab activity is comprised of four parts:

• Part I: Determine the initial launch velocity, $v_{o}$, using measured average range, $\left( \overline{\mathrm{\Delta}x} \right)$, from horizontal launch and known launch height, $\mathrm{\Delta}y$.

• Part II: Measure projectile range (${\mathrm{\Delta}x}_{meas}$) launched at various angles (35^o, 45^o, & 55^o), using calculated initial velocity ($v_{o})$ with associated errors ($\delta(v_{o})$) from Part 1.

• Part III: Calculate ranges (${\mathrm{\Delta}x}_{cal}$) with associated error ($\delta(\mathrm{\Delta}x_{calc})$) for projectiles launched at different angles (35^o, 45^o, & 55^o)

• Part IV: Compare measured ranges ( Δx_meas) to the calculated ranges (${\mathrm{\Delta}x}_{cal}$) using t-statistics.

Part I: Determine initial launch velocity, $\mathbf{v}_{\mathbf{o}}$ using measured range from horizontal launch

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 ($\mathrm{\Delta}x)$ and reference height ($\mathrm{\Delta}y)$ were recorded. The range measurements were repeated 10 times, keeping $\mathrm{\Delta}y$ the same. The average range, $\left( \overline{\mathrm{\Delta}x} \right),$ and corresponding error, $\delta\left( \overline{\mathrm{\Delta}x} \right),$ were then calculated using equations A1.1 through A1.3, from Appendix 1. Table 1 shows a sample of student data; values of $\overline{\mathrm{\Delta}x},\ \delta\left( \overline{\mathrm{\Delta}x} \right),\ \mathrm{\Delta}y,\ and\ \delta(\mathrm{\Delta}y)$ are shown. These measurements were used to calculate the initial launch velocity using equation 4.

Part II: Measuring projectile range ($\mathbf{\mathrm{\Delta}x}_{\mathbf{meas}}$) launched at various angles (35^o, 45^o, & 55^o)

Following the experimental set-up shown in Figure 2, a projectile was launched at three different angles and ${\mathrm{\Delta}x}_{meas}(\theta)$ was measured for each. These ${\mathrm{\Delta}x}_{meas}$ measurements were repeated 10 times for each angle. Table 2 shows sample of student data listing the values of ∆y, (${\overline{\mathrm{\Delta}x}}_{meas})$, and ($\delta(\overline{\mathrm{\Delta}x})$) for the different launch angles. Here, ${\overline{\mathrm{\Delta}x}}_{meas}$, and ($\delta{\overline{\mathrm{\Delta}x}}_{meas})$ denotes the average of the 10 values of ${\mathrm{\Delta}x}_{meas}$ and associated errors, respectively.

Part III: Calculating ranges ($\mathbf{\mathrm{\Delta}x}_{\mathbf{cal}}$), for projectiles launched at different angles (35^o, 45^o, & 55^o)

This part dealt with calculated ranges, ${\mathrm{\Delta}x}_{cal}$, for the projectile launched at the different angles using the initial velocity $v_{o} \pm \delta v_{o} = (2.835 \pm 0.004)\frac{m}{s}$ and the known launch heights from Part I. To calculate ${\mathrm{\Delta}x}_{cal}$, students

1. Determined the initial x- and y-components of the launch velocities ($v_{ox}\ and\ v_{oy})$,

2. Found the time of flight ($t_{cal})$ of the projectile,

3. Calculated, ${\mathrm{\Delta}x}_{cal} = v_{ox} t_{cal}$,

4. and recorded the calculated range in the form:

${\mathrm{\Delta}x}_{cal} + {\delta(\mathrm{\Delta}x}_{cal})$ (8)

Table 3 lists the calculated values of $v_{ox}$ and $v_{oy}$ for the different launch angles along with corresponding errors $\delta (v_{ox})$ and $\delta (v_{oy})$ using equations A2.6 and A2.7. Table 4 lists the time of flight $t_{cal}$ and associated error $\delta (t_{cal})$ for the different angles using equations A2.8 and A2.12. Finally, Table 5 lists the calculated range, $\mathrm{\Delta}x_{cal}(\theta)$, for projectile at different launch angles.

Data Collection and Analysis

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

The above measured values of $\overline{\mathrm{\Delta}x} \pm \delta(\overline{\mathrm{\Delta}x})$ and $\overline{\mathrm{\Delta}y} \pm \delta(\overline{\mathrm{\Delta}y})$ were used to calculate $v_{o}$ along with its error $\delta(v_{o})$ by using equations (4) and (A2.5) and the result is reported as $v_{o} \pm \delta v_{o} = (2.835 \pm 0.004)\frac{m}{s}.$

TABLE 2: Measured ranges for projectile launched at different angles (precision of compass $\delta\theta = {0.5}^{o}$)

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 $v_{ox}\ and\ v_{oy}$ for the different launch angles with associated errors $\delta\left( v_{ox} \right)$ and $\delta\left( v_{oy} \right)$ using A2.6 and A2.7, derived in Appendix 2.

$\delta\left( v_{ox} \right) = \left| v_{o}{cos}\theta \right|\sqrt{\left( \frac{\delta\left( v_{o} \right)}{v_{o}} \right)^{2} + \left( \frac{{(sin}{\theta)} \delta\theta)}{\cos\theta} \right)^{2}}$ (A2.6)

$\delta\left( v_{oy} \right) = \left| v_{o}{sin}\theta \right|\sqrt{\left( \frac{\delta\left( v_{o} \right)}{v_{o}} \right)^{2} + \left( \frac{({cos\ \theta)}{\delta\theta}}{\sin\theta} \right)^{2}}$ (A2.7)

TABLE 3: Initial launch velocities ($v_{ox}\ and\ v_{oy})$ for different launch angles ($\theta)$

Table 4 lists the time of flight $t_{cal}$ and associated error $\delta\left( t_{cal} \right)$ for the different angles using A2.8 and A2.12 derived in Appendix 2, reported below.

$t_{cal} = \frac{\left( v_{o} \sin{\theta \pm}\ \sqrt{\left( v_{o} \sin\theta \right)^{2} + 2 g h}\ \right)}{g}$ (A2.8)

${\delta t}_{cal} = \frac{1}{g}\left\{ v_{o} \sin\theta\sqrt{\left( \frac{\delta v_{o}}{v_{o}} \right)^{2} + {(\cot{\theta \delta\theta)}}^{2} + \frac{{{((sin}{\theta)} \delta v_{o})}^{2}}{{(v_{o} \sin{\theta)}}^{2} + 2gh} + \frac{\left( v_{o}\left( \cos\theta \right) \delta\theta \right)^{2}}{{(v_{o} \sin{\theta)}}^{2} + 2gh}} \right\}$ (A2.12)

TABLE 4: Calculated $t_{cal}\ and,\ \ \delta t_{cal}$ for different launch angles

Table 5 lists the calculated range values (${\mathrm{\Delta}x}_{cal} = v_{ox} t_{cal}$) and associated errors $\delta(\mathrm{\Delta}x_{cal})$ using the values from Tables 4 and 5and equation A2.13 from Appendix 2 for different angles and reported below.

$\delta\left( {\mathrm{\Delta}x}_{cal} \right) = \left| {\mathrm{\Delta}x}_{cal} \right| \sqrt{\left( \frac{\delta v_{0x}}{v_{ox}} \right)^{2}{+ \left( \frac{\delta t_{cal}}{t_{cal}} \right)}^{2}}$ (A2.13)

TABLE 5: Calculated range, ${\mathrm{\Delta}x}_{cal}(\theta$) for projectile at different launch angles

TABLE 6: Measured and calculated ranges, and $p$ values

Part IV: Analysis of measured and calculated ranges ($\mathbf{\mathrm{\Delta}x}_{\mathbf{meas}}\mathbf{\ and\ }\mathbf{\mathrm{\Delta}x}_{\mathbf{cal}}$)

in this part, students compared ${\mathrm{\Delta}x}_{meas}\ and\ {\mathrm{\Delta}x}_{cal}$ by employing t-statistics. In this section, ${\mathrm{\Delta}x}_{m - c}$ is used to denote the absolute value of the difference between the measured and calculated range with corresponding error as ${\mathrm{\delta}({\Delta}x}_{m - c})$. The p-value is then given as $p = \frac{{\mathrm{\Delta}x}_{m - c}}{\delta\left( {\mathrm{\Delta}x}_{m - c} \right)}$, where

${\mathrm{\Delta}x}_{m - c} = \ \left| {\mathrm{\Delta}x}_{meas} - {\mathrm{\Delta}x}_{cal} \right|$ (A1.6)

${\mathrm{\delta}({\Delta}x}_{m - c})=\sqrt{\left( \delta{\mathrm{\Delta}x}_{meas} \right)^{2} + \left( \delta{\mathrm{\Delta}x}_{cal} \right)^{2}}$ (A1.7)

Table 6 shows measured and calculated ranges from Tables 3 and 4, and $p$ values.

Data in Table 2 clearly show that the lunch angle of $\theta = 45^{o}$ does not yield the maximum range. At this point, students were directed to explore, discuss among themselves, and hypothesize why the launch angle of $\theta = 45^{o}$ did not yield the maximum range in this case where the projectile is launched from an initial launch height. This piece of the exercise is explicitly included for students to distinguish between a projectile launched from ground level versus one launched from an initial launch height.

Students were encouraged to explore the meaning of their t-statistics values, especially the ratio of $\left| {\mathrm{\Delta}x}_{meas} - {\mathrm{\Delta}x}_{cal} \right|$ to $\sqrt{\left( \delta{\mathrm{\Delta}x}_{meas} \right)^{2} + \left( \delta{\mathrm{\Delta}x}_{cal} \right)^{2}}$ and relate their results to Table 7 in Appendix 1. Students noted that $p < 1$ indicates no statistical difference between measured and calculated ranges.

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).

Conclusions

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, $v_{o}$ with associated error, $\delta v_{o}$. In Part II, students launched the projectile at various angles, to measure ranges, $\mathrm{\Delta}x_{meas}$. In Part III, students calculated time of flight, $t$ with corresponding errors, $\delta t$, calculated range, $\mathrm{\Delta}x_{cal}$, with associated $\delta(\mathrm{\Delta}x_{cal})$. In Part IV, students compared the measured and calculated values of $\mathrm{\Delta}x$, by using t-statistics. Students reported p-values using equation 9:

$p = \frac{\left| {\mathrm{\Delta}x}_{meas} - {\mathrm{\Delta}x}_{calc} \right|}{\sqrt{\left( \delta{\mathrm{\Delta}x}_{meas} \right)^{2} + \left( \delta{\mathrm{\Delta}x}_{cal} \right)^{2}}}$ (9)

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.

Reflection & Moving Forward

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 $(p < 1)$ between the measured and calculated ranges. Therefore, their measured values were acceptable within their error bounds. Introducing this concept in an introductory course gives students an analytical tool to use in order to critically evaluate their results rather than simply relying on a “gut feeling” that their results are “good” or “bad.”

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.

Appendix 1: Review of Statistics and Error Propagation

This Appendix summarizes simple statistics and error propagation methods.

Review of Statistics

Presented here is a review of finding the mean, standard deviation and associated errors. Suppose we measure a quantity N times (N ≥ 10), then

1. The mean ($\overline{x})$ of the N measurements can be written as:

$\overline{x\ } = \frac{\sum_{1}^{N}x_{i}}{N}$ (A1.1)

The standard deviation σ of the sample measurements can be written as:

$\sigma = \sqrt{(\sum_{1}^{N}{{(x_{i} - \overline{x})}^{2})/(N - 1)\ }}$ (A1.2)

2. The “error” in the measurement is the standard error ($\sigma_{N} = \sigma/\sqrt{N})$ can be written as:

$\sigma_{N} = \frac{\sigma}{\sqrt{N}}$ (A1.3)

3. The mean value of measurement with error is reported as:

$Measurement = \ \overline{x\ } \pm {\ \sigma}_{N}$ (A1.4)

4. The t-statistics to determine whether the difference between two independent values ${\overline{x}}_{1}\ \&\ {\overline{x}}_{2}$ is significant or not can be determined from:

$p = \frac{\overline{\mathrm{\Delta}x}}{\sigma_{\mathrm{\Delta}x}}$ (A1.5)

where,

$\overline{\mathrm{\Delta}x} = \left| {\overline{x}}_{1} - {\overline{x}}_{2}\ \right|$ (A1.6)

and,

$\sigma_{\mathrm{\Delta}x} = \sqrt{\sigma_{x1}^{2} + \sigma_{x2}^{2}}$ (A1.7)

where $\sigma_{x1}$ and $\sigma_{x2}$ are the standard errors associated with ${\overline{x}}_{1}$and ${\overline{x}}_{2}$, respectively.

TABLE 7: Meaning of p values

Rules for Error Propagation

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 $X,\ Y,\ Z,\ \ldots$ are measured values with $\delta X,\ \delta Y,\ \delta Z,\ldots.$ as associated errors, then the following equations apply:

1. Addition

$R\ = \ X\ \pm \ Y;\ \ \ \ \ \ \ \ \ \delta R\ = \ \sqrt{{(\delta X)}^{2} + ({\delta Y)}^{2}}$ (A1.8)

2. Multiplication

$R\ = \ X\ \ Y;\ \ \delta R\ = |R|\sqrt{{(\frac{\delta X}{X})}^{2} + {(\frac{\delta Y}{Y})}^{2}}$ (A1.9)

3. Division

$R\ = \frac{X}{Y}\ ;\ \ \ \ \ \ \ \ \ \ \delta R\ = \ |R|\sqrt{{\left( \frac{\delta X}{X} \right)^{2} + \left( \frac{\delta Y}{Y} \right)^{2}}^{}}$ (A1.10)

4. Multiplying by a constant, $c$

$R = c X;\ \delta R\ = |c| \delta X$ (A1.11)

5. Single-valued functions: For a single-valued function R(X), the associated error $\delta(R(x))$ is:

$\delta\left( R(X) \right) = \left( \frac{dR(X)}{dX} \right)\delta X$ (A1.12)

6. Multi-valued Functions: For a multi-valued function R(X, Y, Z, …), in which X, Y, Z, … are uncorrelated, then:

$R = R(X,Y,\ \ldots.);\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \delta R = \sqrt{{(\frac{\partial R}{\partial X} \delta X)}^{2} + {(\frac{\partial R}{\partial Y} \delta Y)}^{2} + \ldots}$ (A1.13)

7. Polynomial Functions (R based on polynomial function of one variable X)

$R = X^{n};\ \ \delta R = \ |n|X^{n - 1} \delta X\ \mathbf{or\ }\ \delta R = \ |n| \frac{\delta X}{|X|} |R|$ (A1.14)

Appendix 2: Derivations of Equations with their corresponding error

The initial launch velocity and associated error is derived by starting with equation (4). $v_{o}$ can be written as:

$\ v_{o} = \sqrt{\frac{g}{2}} (\mathrm{\Delta}x \frac{1}{\sqrt{\mathrm{\Delta}y}})$ (A2.1)

so,

$\delta v_{o} = \ \sqrt{\frac{g}{2}}\delta F(\mathrm{\Delta}x,\mathrm{\Delta}y)$ (A2.2)

where

$F(\mathrm{\Delta}x,\mathrm{\Delta}y) = (\mathrm{\Delta}x) (\frac{1}{\sqrt{\mathrm{\Delta}y}}))$ (A2.3)

Using equation (A1.9), $\delta F(\mathrm{\Delta}x,\mathrm{\Delta}y) = \sqrt{\left( \frac{\partial F}{\partial\mathrm{\Delta}x} \right)^{2} (\delta\mathrm{\Delta}x)^{2} + \left( \frac{\partial F}{\partial\mathrm{\Delta}y} \right)^{2} (\delta\mathrm{\Delta}y)^{2}}$

with $\frac{\partial F}{\partial(\mathrm{\Delta}x)} = \left( \frac{1}{\sqrt{\mathrm{\Delta}y}} \right)$ and $\frac{\partial F}{\partial(\mathrm{\Delta}y)} = (\mathrm{\Delta}x) \left| - \frac{1}{2}{(\mathrm{\Delta}y)}^{- \frac{3}{2}} \right|$, we can write:

$\delta F(\mathrm{\Delta}x,\mathrm{\Delta}y) = \sqrt{{(\frac{1}{\mathrm{\Delta}y}) (\delta\mathrm{\Delta}x)}^{2} + \frac{1}{4}{(\mathrm{\Delta}x)}^{2} (\mathrm{\Delta}y)^{- 3} {(\delta\mathrm{\Delta}y)}^{2}\ }$ (A2.4)

Therefore,

$\delta v_{o} = \ \sqrt{\frac{g}{2}}\sqrt{{(\frac{1}{\mathrm{\Delta}y}) (\delta\mathrm{\Delta}x)}^{2} + \frac{1}{4}{(\mathrm{\Delta}x)}^{2} (\mathrm{\Delta}y)^{- 3} {(\delta\mathrm{\Delta}y)}^{2}\ }$ (A2.5)

Now, we show how $v_{ox}$ and $v_{ox}$are determined at different launch angles. Let $v_{ox} = v_{o}{cos}\theta = G\left( v_{o},\theta \right)$ and $v_{oy} = v_{o}{sin}\theta = H\left( v_{o},\theta \right)$. Then, applying equation A1.9, we find that:

$\delta\left( v_{ox} \right)=\sqrt{{(\frac{\partial G}{{\partial v}_{o}})}^{2} {(\delta v_{o})}^{2} + \left( \frac{\partial G}{\partial\theta} \right)^{2} {(\delta\theta)}^{2}} = \sqrt{{(\cos\theta)}^{2} {(\delta v_{o})}^{2} + \left( {- v}_{o} \sin\theta \right)^{2} {(\delta\theta)}^{2}}$

$= \left| v_{o}{cos}\theta \right|\sqrt{\left( \frac{\delta\left( v_{o} \right)}{v_{o}} \right)^{2} + \left( \frac{{(sin}{\theta)} \delta\theta)}{\cos\theta} \right)^{2}}$ (A2.6)

Similarly, applying equation (A2.2) to $H\left( v_{o},\theta \right)$, we find,

$\delta\left( v_{oy} \right) = \sqrt{{(\frac{\partial H}{{\partial v}_{o}})}^{2} {(\delta v_{o})}^{2} + \left( \frac{\partial H}{\partial\theta} \right)^{2} {(\delta\theta)}^{2}} = \sqrt{{(\sin\theta)}^{2} {(\delta v_{o})}^{2} + \left( v_{o} \cos\theta \right)^{2} {(\delta\theta)}^{2}}$

$= \left| v_{o}{sin}\theta \right|\sqrt{\left( \frac{\delta\left( v_{o} \right)}{v_{o}} \right)^{2} + \left( \frac{({cos\ \theta)}{\delta\theta}}{\sin\theta} \right)^{2}}$ (A2.7)

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 ($\theta)$ from a reference height (${\Delta}y = \ - h$ ) with coordinates (0, 0) is calculated from: $-h = (v_{o}\sin{\theta)} t - \frac{1}{2}g{t_{cal}}^{2}$ as:

$t_{cal} = \frac{\left( v_{o} \sin{\theta \pm}\ \sqrt{\left( v_{o} \sin\theta \right)^{2} + 2 g h}\ \right)}{g}$ (A2.8)

To find $\delta\left( t_{cal} \right)$, we write equation (A2.8) as:

$t_{cal} = \frac{1}{g} \left\lbrack \delta\left( v_{oy} \right) + U\left( v_{o},\theta \right) \right\rbrack$ (A2.9)

where $U\left( v_{o},\theta \right) = \sqrt{\left( v_{o} \sin\theta \right)^{2} + K}$ and $K = 2gh.$ Applying equation (A1.10) to equation A2.9,

$\delta\left( t_{cal} \right) = \frac{1}{g}\sqrt{{(\delta\left( v_{oy} \right))}^{2} + {(\delta U\left( v_{o},\theta \right))}^{2}}$ (A2.10)

Applying equation A1.9 to $U\left( v_{o},\theta \right)$

$\delta(U\left( v_{o},\theta \right) = \sqrt{{(\frac{(v_{o} \sin{\theta) {(sin}{\theta)} \delta v_{o}}}{\sqrt{{(v_{o} \sin{\theta)}}^{2} + K}})}^{2} + {(\frac{{{(v}_{o}}^{2} \sin{\theta)} \left( \cos\theta \right) \delta\theta}{\sqrt{{(v_{o} sin{\theta)}}^{2} + K}})}^{2}}$ (A2.11)

Substituting equations (A2.7) and (A2.11) in equation (A2.10),

${\delta t}_{cal} = \frac{1}{g}\sqrt{\left( v_{o}{sin}\theta \right)^{2} \left( \left( \frac{\delta\left( v_{o} \right)}{v_{o}} \right)^{2} + \left( \frac{({cos\ \theta)}{\delta\theta}}{\sin\theta} \right)^{2} \right) + \frac{{((v_{o} \sin{\theta) {(sin}{\theta)} \delta v_{o})}}^{2}}{{(v_{o} \sin{\theta)}}^{2} + K} + \frac{{({{(v}_{o}}^{2} \sin{\theta)} \cos\theta) \delta\theta)}^{2}}{{(v_{o} \sin{\theta)}}^{2} + K}}$

${\delta t}_{cal} = \frac{1}{g}\left\lbrack v_{o} \sin\theta\sqrt{\left( \frac{\delta v_{o}}{v_{o}} \right)^{2} + {(\cot{\theta \delta\theta)}}^{2} + \frac{{{((sin}{\theta)} \delta v_{o})}^{2}}{{(v_{o} \sin{\theta)}}^{2} + K} + \frac{\left( v_{o}\left( \cos\theta \right) \delta\theta \right)^{2}}{{(v_{o} \sin{\theta)}}^{2} + K}} \right\rbrack$

${\delta t}_{cal} = \frac{1}{g}\left\{ v_{o} \sin\theta\sqrt{\left( \frac{\delta v_{o}}{v_{o}} \right)^{2} + {(\cot{\theta \delta\theta)}}^{2} + \frac{{{((sin}{\theta)} \delta v_{o})}^{2}}{{(v_{o} \sin{\theta)}}^{2} + 2gh} + \frac{\left( v_{o}\left( \cos\theta \right) \delta\theta \right)^{2}}{{(v_{o} sin{\theta)}}^{2} + 2gh}} \right\}$ (A2.12)

Finally, the derivation of the error in ∆x_cal is derived starting from $\ {\mathrm{\Delta}x}_{cal} = v_{ox} t_{cal}$, to find

$\delta\left( {\mathrm{\Delta}x}_{calculated} \right) = \left| {\mathrm{\Delta}x}_{calculated} \right| \sqrt{\left( \frac{\delta v_{0x}}{v_{ox}} \right)^{2}{+ \left( \frac{\delta t_{cal}}{t_{cal}} \right)}^{2}}$ (A2.13)

Acknowledgements

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.

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