Re: [Phys-l] Teaching Special Relativity

[John Mallinckrodt <ajm@csupomona.edu>]

I don't think or claim that it is a pedagogical disaster to say that  
it might seem "as if" the mass of the object were increasing with  
velocity.  But I do think that it approaches pedagogical malpractice  
not to point out in something like the same breath that it should not  
be taken seriously because it leads to all sorts of other problems.   
Furthermore, it seems to me to be a simple and appropriate matter to  
go on to point out that the more appropriate way to view the result  
is either

1. that the classical formula for momentum is simply wrong although  
it obviously works very well at low speeds

and/or

2. as John Denker has pointed out, that we are simply using the wrong  
velocity, that the correct velocity measures distance traveled in the  
frame of the observer per unit time interval in the frame of the  
object (i.e., the spatial component of the "proper velocity.")

I'd also take this opportunity to say that, just as I firmly believe  
that mass should be viewed as a property of an object, I also agree  
with John Denker that the length of an object should be, and is  
certainly more elegantly viewed as a property of the object (and  
further that the time interval between causally connected events and  
the distance between acausally connected events should be viewed as  
properties of the relationship between the events.)  Nevertheless,  
unlike the case with mass, it seems to me that it is easy to  
operationally define a frame-dependent "length of a moving  
object" (and frame dependent times and distances between events) and  
that there is not, in these cases, such grave downsides to doing so.   
Moreover, I maintain that considerable enlightenment accrues to those  
who follow that path carefully, work through several of the apparent  
paradoxes, come to see that relativity is beautifully self- 
consistent, and eventually graduate to a more elegant worldview based  
on the geometry of spacetime and invariant quantities.

John Mallinckrodt
Cal Poly Pomona

On Jul 3, 2009, at 8:29 AM, Richard Tarara wrote:

> ----- Original Message ----- From: "John Mallinckrodt"  
> <ajm@csupomona.edu>
>
> > > This is what the intro student (indeed myself) comes to special  
> > > relativity with.  The magnetic field example (again I must say  
> > > that  I've calculated and set the fields for
> > > bending high speed protons using 'relativistic mass' and been  
> > > quite successful in getting the beam to the target) presents us  
> > > with a measured velocity but a momentum that has increased non- 
> > > linearly  with that velocity.
> >
> > Of course.  That's what momentum does.  And I'll bet that I can  
> > set  the fields pretty accurately too using the simple fact that  
> > the  required field is directly proportional to the momentum.
>
> OK--then help me out here, for my gen-ed students and in fact  
> myself.  Low velocity momentum is mass x velocity.  We know the  
> velocity of a an object is limited by 'c'.  The momentum of a fast  
> object increases faster than the velocity.  Is there really any  
> 'conceptual' way to view this other than that the mass has  
> effectively increased.  Saying just that the momentum increases non- 
> linearly may be the most 'correct' thing to say, but not very  
> satisfying.  I know JD will say use time-space diagrams and maybe  
> that is best for science, especially physics students, but I don't  
> have the time and I doubt the ability of my gen-ed students to go  
> this route.
>
> BTW:  I once upon a time used Eugene Hecht's gen-ed text "Physics  
> in Perspective" where he clearly talks about mass being velocity  
> dependent but in his Calculus level text he does not take that same  
> approach.  That, IMO, is probably a wise approach overall.
>
> Rick