Collisions¶
About this lesson plan¶
This is a lesson plan for indented for realization during 2h lesson activities.
It has been developed during work in iCSE4school project based on lesson carried out in 2015-2017 at The Stefan Batory High School in Chorzów.
It was prepared by Adam Ogaza based on his lesson.
Attention!
In each of the “code” cells you can change any number, text or instruction. In order to return to the original version refresh the webpage. Sometimes the next code depends on variables defined from the previous one, so one has to execute cells in order of apperance.
Description of the problem¶
The experiment presented in this document has been accomplished during facultative activities and has the status of optional task. Interested pupils might conduct it, film it and describe in a report, additionally evaluated.
The task is to analyze a central collision of 2 bodies of known masses on air truck. The original gliders of air track were equipped with elastic metal bumpers. Because I noticed, that during collisions part of the energy dissipates, causing permanent deformations of bumpers, I replaced them by repelling neodymium magnets. One can expect, that now the collisions will be perfectly elastic, so that both momentum and kinetic energy will be conserved. Such experiment gives also opportunity for detailed investigation the nature of magnetic forces.
Collisions lasted about 3 seconds and were filmed using HD cameras. Exemplary film is available here: https://youtu.be/JSpKctrX_YM
The pupil’s task was to read out (frame by frame) positions of both gliders from the air track’s scale. There were a few gliders to choose from, each of different mass. The variables \(x_1\) and \(x_2\) denote coordinates of the ends of gliders’ grounds turned to each other. The magnets glued to gliders jutted by \(d_1\) and \(d_2\), so that the distance between poles was less from \(|x_2-x_1|\) by \(d_1+d_2\). The set of experimental data was the following:
Gliders’ masses: \(m_1\) and \(m_2\) from about 0,17 kg to over 0,4 kg
Sizes of magnets and their fastening: \(d_1\) and \(d_2\), about 0,01 m
Time was calculated iteratively and, depending on the camera, was increasing with the step 1/25 s or 1/30 s
Gliders’ coordinates \(x_1(t)\) i \(x_2(t)\) red out from the film and expressed in meters. If gliders were moving towards decreasing values on the scale, their velocities and momentums were negative.
On the basis of above data, one should draw and interpret the following charts:
\(x_1(t)\) and \(x_2(t)\) - gliders’ positions as a function of time.
\(v_1(t)\) and \(v_2(t)\) - gliders’ instantaneous velocities in particular frames. When points of charts created a chaotic cloud (data noising), I advised to take averages from 2 neighbor frames.
\(p_1(t), p_2(t)\) and \(p_c(t)\) - momentums of particular gliders and the whole system. One should check, whether the total momentum was conserved.
\(E_1(t), E_2(t)\) and \(E_c(t)\) - gliders’ kinetic energies and their sum. One should check, whether the total kinetic energy was conserved.
\(a_1(t)\) and \(a_2(t)\) - gliders’ accelerations.
\(F_1(t)\) and \(F_2(t)\) - forces applied to gliders.
\(F(r)\) - dependence of magnetic force on the distance between poles.
Remarks concerning implementation¶
Data of first student
It was a trailblazing performance of this experiment in April 2016, done by a single volunteer from the firs age-group of pupils involved in the project.
Results of measurements and charts \(x_1(t)\) and \(x_2(t)\):
Charts \(v_1(t)\) and \(v_2(t)\):
We see a big noise caused by a limited resolution in reading out the coordinates in time. A small fluctuations of position increases in respective frames, invisible on position charts, are enough to spill the chart of velocity. The data noise devolves (and increases) into charts of momentums, kinetic energies (here we have a square of speed), accelerations and forces. I advised to take average speed for every two neighbor frames:
The noise decreased, but it refers also to the amount of chart points and time resolution of investigated phenomenon. We lose the most interesting processes going on in a short while during the closest gliders’ approach.
The pupil made the rest of charts, wrote the report and drew conclusions, but in the cloud of measurement points it was difficult to discern any interesting details. The noise can be reduced by advanced mathematical means, far away from high school pupil’s capabilities.
Data of second student
In October 2016 the whole group of next pupils age-group filmed their collisions. They tried to investigate different cases, varying the gliders’ masses, values and senses of velocity or putting one glider motionless (as a target). I dish up data from the best elaboration (referring to film quoted above).
Results of measurements and charts \(x_1(t)\) and \(x_2(t)\)
Charts \(v_1(t)\) and \(v_2(t)\)
The averaging over neighbor frames was applied immediately, to reduce the noise.
I will not depict the way of creating further charts, because from the IT point of view there is nothing revelatory in it. Pupils are able to write the proper code by themselves. It is enough to know, how loops work, how to create charts and fit functions to measurement points. In case of doubts, the original student’s homework will help:
Polish version: https://sage01.icse.us.edu.pl/home/pub/146/
English version: https://sage01.icse.us.edu.pl/home/pub/147/
I will add only, that I do not share all the presented there final conclusions. Furthermore, in my opinion, the author quite unnecessary fitted 12th degree polynomials to the charts of velocity. I do not know, what was it for.
Conclusions¶
The presented experiment was one of most interesting in my professional career. Because of the multiplicity of different situations (arbitrary velocities, more gliders to choose, one could obtain quite different results. Students had freedom in drawing conclusions, it was their independed research work. For example, in the quoted work, at the chart of kinetic energy there is an evident minimum at the moment of closest approach of gliders. Student interpretent it as a measuring error caused by too rapid change of speeds. In my opinion, it is a moment, when kinetic energy partially remolded into energy of magnetic interactions. But why, in such case, there is also a visible small breakdown at the total momentum chart?
This experiment basically pertained to pure mechanics, but by the way it gave an opportunity to explore the nature of magnetic force. Students could fit any curves to the chart \(F(r)\) - I did not impose ready solutions. One should look at the data and figure out, what type of curve will be the most suitable.
How it can be seen from above analysis, the precision in reading out the positions of gliders at individual frames, is essential. It is not easy and requires use of good camera. The data noise may be partially removed, but it decreases the resolution, in which we see the whole phenomenon.
The added value of the whole enterprise was a fruitful cooperation with English teachers. They supervised translations done by authors of the best homeworks. Students in science profile classes realize an additional subject called English for engineers. Experiment in physics created opportunity to exercise technical language on living example and get additional assessment for it.
