M and I: chapter 1 - Relativistic vs. Classical Momentum

In my last post, I promised to share thoughts and materials.  Here's the first!

Chapter 1 is a lot of review for my second-year students. The new bit is relativistic momentum, and the revelation that our definition of momentum from last year was really just a special-case limit.

One of my themes for the year is 'pulling back the veils' on all of those more complicated bits that we approximated and hand-waved away last year. I want this to be a major selling point of the course. Unfortunately, as presented, I think that the momentum definition isn't accessible or real to the kids.  We can't really discover it, and it's just an abstraction at first.

I wrote a VPython program to give this some context. You can run it twice for the students - in each case, a 60 g block is pushed from rest by a constant 1 million Newton force. The momentum and velocity of the block are graphed as time goes on.

Here's where another pillar of the course is going to come in to play: fundamental laws of the universe. Really, we didn't deal with too many last year, at least not on a really visceral and important level. Here, though, the universal speed limit becomes important. Look at the v graph pass right by it without blinking, when the program uses the classical definition of p (and finds v from there):

That's a problem. The problem's with our model, though: momentum just doesn't really equal mass times velocity.  Uncomment the gamma definition in the program to use the real relationship now:

Here's a great way to distinguish which is the fundamental principle here: the momentum principle still holds just fine, even in the relativistic case, but our definition of v (as determined from the momentum) was broken. With the relativistic relationship, the velocity approaches the universal speed limit.  Two days in and kids not only know a fundamental law of the universe, but they know how it really does fit in with what they knew about momentum and what the missing piece in their model was.

Script:

 from __future__ import division, print_function  
 from visual import *  
 from visual.graph import *  
 scene.background = color.white  
 scene.height = 50  
 scene.width = 50  
 scene.x = scene.y =0  
 ## Josh Gates 2012, started with kernel of Ruth Chabay program  
 print ("""  
 Click anywhere in display window to start.  
 Click to stop.  
 Momentum and velocity are shown as .6 kg block is pushed with a constant 1,000,000 N force.  
 Uncomment gamma line to remove classical p def'n/assumption""")  
   
 ## CONSTANTS  
 delta_t = 0.01  ## for 100 steps  
 mblock = 0.06  
 c=3e8  
 Fnet = vector(1e6,0,0)  
   
 ## OBJECTS & INITIAL VALUES  
 pblock = mblock*vector(0,0,0)  
 gamma = 1  
   
 # start time at 0  
 t = 0  
 scene.center = (0,.1,0)  # move camera up  
 scene.range = 0.15  
   
 ## GRAPH STUFF  
 gp = gdisplay(background=color.white, foreground=color.black, y=0, x=250, height=300,  
        title='Momentum vs. time: block with constant F', xtitle='Time (s)', ytitle='Position (m)',  
        ymax=3e7,xmax=30)  
 blockp = gcurve(color=color.magenta)  
 blockp.plot(pos=(t,pblock.x)) ## initial pos.x of block  
   
 gv = gdisplay(background=color.white, foreground=color.black, y=300, x=250, height=300,  
        title='Velocity vs. time: block with constant F', xtitle='Time (s)', ytitle='Velocity (m/s)',  
        ymax = 3e8, ymin=0, xmax=30)  
 blockv = gcurve(color=color.blue)  
 blockv.plot(pos=(t,pblock.x/(gamma*mblock))) #initial velocity  
 asymptote=gcurve(color=color.red)  
   
   
 scene.mouse.getclick()  
 stopper='go'  
   
 a=.1  
 while a<1000:  
   asymptote.plot(pos=(30/a,3e8))  
   a=a+1  
   
 ## CALCULATIONS  
 while stopper=='go':  
   rate(500)   
   
   pblock = pblock + Fnet*delta_t  
   
   # comment to make classical approximation  
   #gamma = (1-(pblock.x)**2/(mblock**2 * c**2+(pblock.x)**2))**-.5  
     
   # update time  
   t = t + delta_t  
   # plot pos.x, velocity.x of block  
   blockp.plot(pos=(t,pblock.x))  
   blockv.plot(pos=(t,pblock.x/(gamma*mblock)))  
 ##  print t, block.y  
   if scene.mouse.clicked:  
     stopper='stop'