{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Programmering endrer fagene\n", "Denne økta handler om hvordan programmering kan endre fagene når vi ikke lenger er begrenset av hva som er analytisk løsbart eller elevenes matematiske kompetanse. \n", "\n", "Vi starter med å se på mekanikken, men vi kunne like gjerne hentet eksempler fra andre fagområder. I ungdomsskolen lærer elevene om sammenhengen mellom posisjon, fart og akselerasjon. Der lærer de også om krefter og energi. Men det er svært begrenset hva de kan se på av virkelighetsnære problemstillinger. \n", "\n", "I Fysikk 1 på videregående skole tas mekanikken litt lenger og en enkel modell for friksjon blir introdusert. Likevel er det begrenset hva elevene kan modellere pga manglende matematikkompetanse. \n", "\n", "## Fallende muffinsformer\n", "La oss se på fall med luftmotstand. Mange lærere har nok sluppet en og to muffinsformer for å illustrere at luftmotstanden øker med kvadratet av farten. En algebraisk øvelse på dette gir oss at hvis vi holder en muffinsform i en gitt høyde $H$ over bakken og så to oppi hverandre i høyden $\\sqrt{2}\\cdot{H}$, så vil de to muffinsformene treffe bakken på likt. \n", "\n", "Men hva om vi vil studere hele bevegelsen og plotte f.eks. tid og fart? Vi kunne bruke en bevegelsessensor til å logge posisjon som funksjon av tid når muffinsformene faller, og så forsøke å lage en modell som beskriver bevegelsen.\n", "\n", "Med Newtons andre lov og kunnskap om sammenhengen mellom posisjon, fart og akselerasjon har elevene nok kompetanse til å programmere dette for å finne en løsning. \n", "\n", "### Underveisoppgave\n", "1. Lag et fri-legeme diagram av en fallende muffinsform (med luftmostand!). Vi setter tyngden $G=mg$ og luftmotstanden $L=kv^{2}$.\n", "2. Bruk Newtons andre lov og forklar at denne gir oss følgende differensiallikning: \n", "$\\frac{dv}{dt}=g-\\frac{k}{m}v^2.$ Hvordan kan vi bruke differensialligningen til å regne ut aksellerasjonen ved gitt tidspunkt når hastigheten er gitt?\n", "3. Forklar hvorfor vi kan skrive $v_{i+1}=v_{i}+a_{i}\\cdot{dt} $ når vi tar små tidssteg. Her er $v_i$ tilnærmet hastighet ved tidspunkt $t_i$ og $v_{i+1}$ tilnærmet hastighet ved tidspunkt $t_{i+1}$.\n", "\n", "Vi prøver oss på selve programmeringen! " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "#Fallende muffinsformer\n", "\n", "\n", "from pylab import *\n", "\n", "#Fysiske størrelser\n", "g=9.81 #tyngdeakselerasjon i m/s/s\n", "m=0.5 #masse til gjenstand som faller i kg\n", "k=0.2 #luftmotstandskoeffisient i Nsmm\n", "\n", "#Tidsintervaller\n", "N=100000 #antall intervaller\n", "tid=2 #antall sekunder\n", "dt=tid/(N)\n", "\n", "#Vektorer/arrays\n", "a=zeros(N) #akselerasjon i m/s/s\n", "v=zeros(N) #fart i m/S\n", "s=zeros(N) #posisjon i m\n", "t=zeros(N) #tid i sekunder\n", "\n", "#Initialbetingelser\n", "v[0]=0 #startfart 0 m/s\n", "s[0]=0 #startposisjon 0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Husk å lage oversiktelige og lesbare programmer - både med tanke på deg selv og elevene. \n", "\n", "Nå har vi lagt klart arrayer med 100000 punkter i hver for både akselerasjon, fart, posisjon og tid. Det neste blir å fylle disse arrayene med tallverdier basert på difflikningen. Det gjør vi med en for-løkke. " ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "#Eulers metode\n", "for i in range(N-1):\n", " a[i]=g-(k/m)*v[i]**2 #utledet fra Newtons 2. lov\n", " v[i+1]=v[i]+a[i]*dt\n", " s[i+1]=s[i]+v[i]*dt\n", " t[i+1]=t[i]+dt\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Underveisoppgave\n", "1. Hvorfor bruker vi en for-løkke og ikke en while-løkke her?\n", "2. Hvorfor går løkken til N-1 og ikke til N?\n", "3. Lag en verditabell for a, v og t for de tre første rundene i løkken. Fyll inn tall i tabellen slik at du skriver med ulik farge avhengig av hvilken runde det er i løkken. Beskriv det du ser. \n", "4. Hva blir verdien til akselerasjonen i det siste punktet? Kan du fikse på programmet ditt så dette blir løst?\n", "\n", "\n", "Nå skal vi plotte resultatet. Først velger vi å plotte både akselerasjonsgraf, fartsgraf og veigraf. Du ser at det kommer litt oppi hverandre. Gjør endringer i programmet slik du ønsker for å få et penere resultat. Husk at du kan kommentere ut hele deler av programmet ved å lage \"\"\" før og etter de linjene du vil kommentere ut. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'\\nNå har jeg kommentert \\nut alle disse\\nlinjene \\n'" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Plotting\n", "subplot(3,1,1)\n", "plot(t,a,'b-')\n", "title('Akselerasjon')\n", "xlabel('tid/s')\n", "ylabel('akselerasjon/(m/s/s)')\n", "grid()\n", "\n", "subplot(3,1,2)\n", "plot(t,v,'r-')\n", "title('Fart')\n", "xlabel('tid/s')\n", "ylabel('fart/(m/s)')\n", "grid()\n", "\n", "subplot(3,1,3)\n", "plot(t,s,'g-')\n", "title('Posisjon')\n", "xlabel('tid/s')\n", "ylabel('posisjon/m')\n", "grid()\n", "tight_layout()\n", "\n", "\"\"\"\n", "Nå har jeg kommentert \n", "ut alle disse\n", "linjene \n", "\"\"\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Litt om modellering\n", "Med dette lille programmet åpner det seg opp mange nye muligheter for å utforske naturvitenskapelige problemstillinger på en ny og dypere måte. \n", "Elevene kan selv undersøke betydningen av de ulike fysiske størrelsene. Hvor raskt oppnår gjenstanden terminalfart når massen endrer seg? Hva med koeffisienten $k$? Kan vi si noe om denne for ulike gjenstander? Og hva om vi var på månen eller en annen planet? Ved å endre på de fysiske størrelsene i begynnelsen av programmet kan datamaskinen regne ut 100 000 punkter for tid, posisjon, fart og akselerasjon på praktisk talt null tid. Dermed kan elevene bruke mer tid til å bygge en intuisjon for fysikkfaget. \n", "\n", "Vi er heller ikke begrenset av modellen $L=kv^2$. Vi kan lett undersøke helt andre modeller som ikke gir løsbare differensiallikninger, men som kan løses numerisk. Her er det bare fantasien som setter grenser :)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#Fart på jorda eller på månen?\n", "\n", "#Jorda\n", "g = 9.8\n", "\n", "#Eulers metode\n", "for i in range(N-1):\n", " a[i]=g-k*v[i]**2/m #utledet fra Newtons 2. lov\n", " v[i+1]=v[i]+a[i]*dt\n", " s[i+1]=s[i]+v[i]*dt\n", " t[i+1]=t[i]+dt\n", "\n", "plot(t,v,'r-')\n", "title('Fart på jorda')\n", "xlabel('tid/s')\n", "ylabel('fart/(m/s)')\n", "grid()\n", "show()\n", "\n", "#Månen\n", "g = 1.6\n", "\n", "#Eulers metode\n", "for i in range(N-1):\n", " a[i]=g-k*v[i]**2/m #utledet fra Newtons 2. lov\n", " v[i+1]=v[i]+a[i]*dt\n", " s[i+1]=s[i]+v[i]*dt\n", " t[i+1]=t[i]+dt\n", " \n", "plot(t,v,'r-')\n", "title('Fart på månen')\n", "xlabel('tid/s')\n", "ylabel('fart/(m/s)')\n", "grid() \n", "show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Underveisoppgave\n", "Finn ut hvor lang tid det tar før gjenstanden når terminalfart på månen. \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.2" } }, "nbformat": 4, "nbformat_minor": 2 }