Effect of air resistance on a baseball: Write a computational model using VPytho

Question

Effect of air resistance on a baseball: Write a computational model using VPython to predict the motion of a baseball that is hit at a speed of 44 m/s at an angle of 45 degrees to the horizontal. A baseball has a mass of 155 g and a diameter of 7 cm. The drag coefficient C for a baseball is about 0.35, and the density of air at sea level is about 1.3 x 10^-3 g/cm^3. In your initial model, neglect air resistance. How far does the ball go? Is this distance reasonable? (A baseball field is about 400ft from home plate to the fence in center field. An outfielder cannot throw a baseball in the air from the fence to home plate.)Plot a graph of K+Ug vs. time for the system of baseball plus Earth. Now add air resistance to your model. How far does the ball go? Compare the graphs of K + Ug with and without air resitance. In Denver, a mile above sea level, the air is about 83 % as dense as the air at sea level. Including the ffect of air resistance, use your computer model to predict the trajectory and range of a baseball thrown in Denver. HOw does the predicted range compare with the predicted range at sea level?

Explanation / Answer

27 Table 8: Codes validated in previous work used for the proctored assignment [59]. Codes marked with as terisks(*) were not applicable to the proctored assignment. GT students who were tasked with modeling the motion of planet s became con- fused when trying to select the appropriate exponen t of the interaction constant [59]. This issue does not arise when model- ing projectile motion and thus IC5 was not needed. In orbital motion, an explicit separation vector is required due to the nature of motion (i.e., where is the separation ve ctor and not necessarily zero), GT students attempt ed to raise this to a power (when in fact it is the magnitude of the diff erence that should be raised not the vector) [59]. In projectile motion the separation vector is always zero therefore there is never a need to interact with it (this is the excl uded FC3). Due to the sim- plicity of the forces involved in the proctored ass ignment students didn't have any other force direct ion confusion (the only force direction would be the direction of gravity) thus FC5 was inapplicable. Using Euler-Cromer integ ration to represent Newton's 2nd law programmatically, students modelin g orbital motion often had difficulties differentia ting between when to use a scalar or a vector [59]. Since the acceler ation is one dimensional in projectile motion SL3 w as not needed. SL4 was in regards to momentum updates and the proctored assig nment does not consider momentum in its calculation s (students were told force is not the more general ) [59]. In projectile motion, the "massive particle" is the E arth. Generally, Newton's theory of gravitation will work for this situation . However this can be reduced to for projectile mo tion and this is done for the proctored assignment removing the need for O1. Code Description IC1 Student used all the correct given values from the grading case IC2 Student used all the correct values from the test c ase IC3 Student used the correct integration time from either the gradin g case or test case. IC4 Student used mixed initial conditions IC5* Students confused the exponents on the units of the exponent of k (interaction constant). FC1 The force calculation was correct FC2 The force calculation was incorrect, but the calcu lation procedure was evident. FC3* The student attempted to raise the sepa ration ve c tor to a power FC4 The direction of the force was reversed SL1 Newton’s second law was correct SL2 Newton’s second law was incorrect but in a form that updates SL3* Newton's second law was incorrect and the stu dent attempted to update it with a scalar force SL4* Student created a new variable for p_f (momen tum) O1* Student attempted to update force, momentum, or pos iti on for the massive particle. O2 Student did not attempt the problem O3 Student did not print final results O4 Coding Error [60] SUG1 Student didn't change anything inside the while loo p SUG2 Student created extra while loops SUG3 Student printed in while loop SUG4 Student did not indicate Newton's 2nd law causes a change in m o- tion Several of the codes needed to be dropped because t hey didn't apply, specifically, some issues were not seen due to the high school students havin g an easier assessment problem. The Georgia Tech (GT) students modeled the motion of orbiting planet s, the high school students modeled the motion of a 28 baseball flying through the air. Two of these codes deal explicitly with vector issues that only arise with complex problems like orbital motion (FC3, SL3). GT students were required to model momentum which was not required for high school students, thus O1 was jettisoned. Finally, the GT students needed to define the exponent of the separation constant whic h is not necessary for projectile motion (code IC5 and FC3). The process for applying these codes had multiple s teps and was based on producing high corre- lation across two coders. Working with another grad uate student, a small selection of proctored as- signments were first analyzed using the codes from previous Georgia Tech work [59]. Examples and def- initions of each code were developed from a subset of the high school student coding assignments. The- se codes were applied to the entire data set. Cross correlation between codes at this stage indicated a low agreement across all of the data set so the sma ller data set selection was revisited. This process was repeated until a high agreement between coders was reached. This process also generated new codes labeled SUG# in Table 8. SUG1-4 all arose from how students wrote code within the while loop. SUG1-3 are computational idiosyncrasies that were interest ing and merit further exploration. SUG4 is interest ing because students were explicitly instructed to defi ne accelerations in their code in terms of force. T hus SUG4 happens when students eschew this for simply w riting the acceleration, in this case 2 m 9.8 s g = or 2 0, 9.8 s , 0 m g = . To calculate the agreement between coders we tabula ted the total number of agreements we had (that is, the total number of times that two co ders agreed that a code applied). Then we counted the number of total codes. For example, if research er A gave three unique codes to a student, and re- searcher B gave two unique codes to a student, and researcher B's codes were contained within re- searcher A's codes than they would disagree 1/3 of the time. Likewise, if researcher A gave two unique codes and researcher B gave three unique codes but they only agreed once, the student would receive 29 four unique codes total and the researchers disagre ed twice. The ratio of the total number of agree- ments to the total number of unique codes gives the fractional agreement (that is, the inter-rater rel ia- bility). In this case researchers agreed 83.8% of t he time [61]. Our analysis of student python code suggests that h igh school students can engage in computa- tional modeling in the context of physics and that these students are generally capable of using numer i- cal computation to solve physics problems. Based on the output of their computational models, stu- dents were placed into one of three categories: “co rrect results and animation” (N = 10, 31%), “pro- duced animation, but incorrect results” (N=8, 25%), and “produced no animation” (N=14, 44%). The group that produced an animation but had incorrect results could be broken down into two groups. They either produced some number of errors either w riting the integration algorithm alone (25%) or writing the integration algorithm and assigning ini tial conditions (75%). The remaining 44% who pro- duced no animation were split into two groups. They either had small syntactic errors (36%) or had nu- merous physics and computational errors (64%). The group that had small syntactic errors most often would have succeeded with minimal input from peers or instructors because these errors were minute (e.g., missing a colon at end of while loop declara tion but the rest of the program is correct). The success rate here is important to note for seve ral factors. First, the students had only spent the fall semester learning to use computation. Duri ng this semester students only had two full activit ies that had computation integrated. They then had a mo nth long holiday break where they did not partici- pate in any classroom activities. Upon returning th e students had two weeks of beginning of year exam- inations that did not include any material on compu tation. After this two weeks the students had a sho rt refresher (<1 day) on the Fall semester's material and then participated in the proctored assignment. Thus students, had at least a 6 week gap between th e education period (i.e., the fall semester where they were learning to use computation in the contex t of force and motion) and the testing period. Four of these weeks were not spent participating in educ ational instruction at all. Furthermore, in an info rmal 30 survey given by the classroom teacher indicates mos t students feel comfortable with the computer but have no programming experience with the computer. T hus, high school students with little program- ming experience can spend a relatively short period of time learning computation and a sizeable fracti on will be able to apply this knowledge in context at a later time. With more instruction and more integr a- tion, a more favorable portion of the class could c omplete this assessment successfully. But do these students who complete code successfully understand it's link to the physics they are modeling? Or have they simply memorized algorithms that can only be u sed in very specific contexts. 2.3 Essay questions The code that students wrote for the proctored assi gnment demonstrated a variety of possible outcomes, but this assessment was unable to probe d eeply how students constructed these computa- tional models. To look more deeply we asked the stu dents to write a short essay. Students responded to this essay question after they completed the pro ctored assignment. The essay question investigated whether students’ success was predicated on simply reproducing an algorithm, or whether successful students made deeper connections between the physic s and the computational algorithm. That is, did these students engage in the practice of computatio nal thinking while developing their computational model? The question given to the students probed th e student understanding of how the computer in- teracts with Newton's 2nd law via the Euler-Cromer integration. Twenty-nine of 32 students completed the essay question. Students could run a working ve rsion of the program before answering the essay question. The practice of computational thinking requires a l ogical problem solving approach that often involves thinking iteratively [59]. To further inve stigate how students developed their computational models, we asked students to describe the integrati on (while) loop mathematically, physically, and pro - grammatically. Specifically, the students were aske d, "Download and run the completed baseball.py program. What is the purpose of the loop? How can y ou describe (mathematically or physically) what 31 the loop does in your program? Run the program agai n and explain what the loop is doing while your program is running." In order to provide a complete explanation, students needed to comment on the iterative procedure of the loop itself and its rela tionship to the integration of the equations of mot ion by the incremental stepping of Newton’s Second Law. Three researchers catalogued all views that "arose" from the student responses. These views formed codes that were tested against a small subse t (<5) of essays. This process was repeated until a n agreement was formed on which codes applied where w ithin the subset of essays. These codes were then applied to all of the submitted essays. This p rocess was repeated until there was a high agreemen t between coders (100%). Figure 5: Students who displayed Force-casual and I terative views were more likely to be successful on the proctored as- signment (N=29). The bars represent a total proport ion of the class (that is out of 100%). The percent ages in this graph are 32 different than the proctored assignment because the number of students who responded to the essays is slightly less (proc- tored assignment N=32). The explanations presented by students in response to this question were captured by four dis- tinct but not necessarily exclusive views. Some (38 %) students presented a “force-causal” view of the loop structure. This view was characterized by a cl ear connection between force and motion. A student presenting a force-causal view would describe how t he force of gravity causes a change in the motion o f the ball; “The loop is constantly changing the velocity of the ball while the Fnet [n et force] stays con- stant . [The force] makes the ball fall faster with every loop that runs”. Another group (17%) of students presented a “kinematic-observational” view of the l oop structure. These students indicated they had observed an acceleration (or some change in a kinem atic quantity), but these students did not connect this observation back to the concept of a non-zero net force. One student with a kinematic- observational view noted, “The loop's purpose is to use the acceleration of the ball to affect the ball's velocity and position. The loop is run every .01 seconds (deltat). It re- updates the velocity and position of the ball at that interval.” Almost two-thirds (65%) of students described the integration loop as a lo cal, iterative process governed by instantaneous influen ces. This iterative-local view was characterized by a discussion of incremental steps through the loop an d statements such as “in this program, the [integra - tion ] loop is what the computer runs through to [compute] a new position, velocity, and all other forces for every [time it executes] .” All the students who exhibited a force-causal vi ew and nearly all students who presented a kinematic-observational view of mot ion also exhibited an iterative-local view of mo- tion. Slightly more than a quarter (28%) of all res pondents fell into no category. This group of stude nts most often wrote very short, incomplete responses t hat were too difficult to classify. More examples of student essay coding can be found in Table 9. 33 Table 9:Student views with examples. Underlined por tions indicate sections coded as applying to the as sociated view. Exam- ples might include multiple views but this in not i ncluded in these examples. Student View Example Iterative procedures The program goes through all the steps and prints/runs them while the "while" statement is still true . While the program is running the loop is going through the steps very fast. Force - Causal (i.e., mentions force and motion quantities together) In this program the loop is what the computer runs through to put in a new position, velocity and all other forces for every .01 seconds. It is similar to a movie in that in a movie there are a bunch of pic- tures that scroll so quickly it looks like it is mo ving . When it is running it moves the ball and updates the time . Kinematic - ob servational The loop shows the changes in every vector as the time changes . I didn't really finish the vpython module so I don't completely understand every- thing yet. It also labels where the ball is at each second. No category No example We compared the views that students presented on th e essay question to their performance on proctored coding assignment. Students with each vie w were binned into the broad proctored assign- ment categories (i.e., “correct results and animati on”, “produced animation, but incorrect results”, a nd “produced no animation”). Students who presented bo th an iterative-local and force-causal view were most likely to produce a correct program. Students whose essay were short and incomplete were most likely to write programs that produced no animation s. Figure 5 summarizes our findings. 2.4 High School Student Interviews Students’ essay responses elucidated that the conce pts of force, motion, and iterative processes should be connected to facilitate computational thi nking. However, investigating how students make these connections requires observing and questionin g students while they engage in the practice of computational thinking. Several weeks after student s completed the essay question, we interviewed five of them while they filled in the missing pieces of a scaffolded computational modeling program on pa- per. During the interview, students also answered q uestions about how they define a force and how 34 forces, motion, and the integration loop are relate d. Students were asked to speak out loud while com- pleting the scaffolded code and answering questions ; their responses were videotaped. Only students whose proctored assignment code produced animations (i.e., “correct results and animation” and “pro- duced animation, but incorrect results”) were invit ed to the study. Six students were chosen to partic i- pate; five completed the interview. Of the students who completed the interview, 3 presented force- causal and iterative-local views on the essay quest ion. One student had previously presented both a ki n- ematic-observational and an iterative-local view, b ut expressed a force-causal and an iterative-local view in the interview. The last student presented a prim arily iterative-local view on the essay question an d in the interview. For students who developed a correct computational model, the interviews further highlighted the links they made between force, motion, and iter ative processes. A student who wrote a correct pro- gram described her code with a force-causal and an iterative-local view, “To predict the velocity you would have to do baseball.v = initial velocity of the baseball pl us gravity times time . That would give me the new velocity after [the execution of] every single loop . And then you need to update the position based on the loop .” This student presents the basic concepts behind Newton’s 2nd law but also de- scribes how the numerical integration loop updates the velocity of the ball in each execution. By con- trast, a student who constructed a model that produ ced incorrect animation demonstrated an incorrect conception of force and motion, “ force generally [is] acquired through motion . There's always force act- ing on an object.” When questioned about how the loop models the physics of the system, the student presented solely an iterative-local view, “[the loop] has formulas that it solves for , like, update position equals [ baseball.pos + baseball.v*deltat ].” While this student was able to generate a computational model for the proctored assignment th at ran without (syntactic) errors, she did not use the correct physics to do so.

Example: r o m v i s u a l . g r a p h i m p o r t v0 = 2 2 . a n g l e = 3 4 . g = 9 . 8 k f = 0 . 9 N = 25 v0x = v0 c o s ( a n g l e p i / 1 8 0 . ) v0y = v0 s i n ( a n g l e p i / 1 8 0 . ) T = 2 . v0y / g H = v0y v0y / ( 2 . g ) R = 2 . v0x v0y / g g r a p h 1 = g d i s p l a y ( t i t l e = ’Projectile with (red)/without (yellow) Air Resistance’ , x t i t l e = ’x’ , y t i t l e = ’y’ , xmax=R , xmin= vx = vx vx = vx e f p l o t A n a l y t i c ( ) : v0x = v0 c o s ( a n g l e p i / 1 8 0 . ) v0y = v0 s i n ( a n g l e p i / 1 8 0 . ) d t = 2 . v0y / g / N p r i n t ( " FRICTIONLESS " ) p r i n t ( " x y" ) f o r i i n r a n g e (N) : r a t e ( 3 0 ) t = i d t x = v0x t y = v0y t

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