Re: [Phys-L] [SPAM] Re: The Make-Believe World of Real World Physics

[John Denker <jsd@av8n.com>]

On 07/29/2013 06:40 PM, Paul Lulai wrote:
> I think this question (how fast / far to have a certain avg speed) is
> a lot of work that isnt worth the payoff and is also a bit of a
> trap. It feels like we are baiting them into trying to do it
> incorrectly. I am sure there are science & technical folks that do a
> lot with average speed, but we do next to nothing with it in high
> school, except ask trick questions like this one. I suppose it is a
> way to check the difference btn speed and the vector nature of
> velocity. It just seems like there would be other ways that could
> check the same conceptual point and be less misleading. I don't do
> much at all w avg speed. While I am sure it can be helpful in some
> situations, this isnt an area I am going to go crazy preparing for.

Several follow-up points:

1) This is a moving target:

 1a) In the early part of the introductory high-school 
  course, there are lots of good reasons to stay away 
  from these "inverse speed" problems.

 1b) OTOH suppose it's a college course.  The students are
  seeing all this stuff for the second time, and they know
  a little bit of calculus.  Then you could make a pretty
  good case for including such problems.  These have some
  direct real-world applications.

 Example:  Suppose you are riding your bike on a hilly 
 course.  It's half uphill and half downhill.  
   Strategy A:  With ordinary effort, you can go 12 mph 
 on the uphills and 24 mph on the downhills.
   Strategy B:  You realize that with extra effort, you 
 can go 1 mph faster (13 mph) on the uphills.  However,
 at this point you are limited by your cardiopulmonary
 capacity, so the only way you can sustain this is to
 take it easy on the downhills, so you go 3 mph slower
 (21 mph).
 
 So, the question arises, is strategy B faster than 
 strategy A?  Is it even remotely possible that gaining
 1 mph uphill is worth losing 3 mph downhill?

 The further question arises:  How do we understand this?

============================

2) This is a fine illustration of the point I have been
 making: if you don't have the tools to answer this question,
 it is indeed hard and scary and counterintuitive.  On the
 other hand, if you do have the tools, it's no big deal.

 Furthermore, the tools that you need to deal with this
 question are tools that you want to learn anyway, because
 they have many other applications.

 We agree that leading students into a trap is a monumentally
 bad idea.  It is better to teach them how to swim /before/
 throwing them into deep water.

 Hint:  I like to think in terms of /weighted averaging/.
  *) If you are averaging over time, you want the average
   of the velocity.
  *) If you are averaging over distance, you want the
   average over the _inverse velocity_.
   -- Note that taking the inverse is a nonlinear operation,
    and it does _not_ commute with taking the average.

 Further hint:  You can understand the effect of small 
 /changes/ in speed by expanding things to first order.

 The students grew up on a steady diet of simple linear
 problems.  This is fine as a starting point, but not as
 an ending point.  Real life is highly nonlinear.  At
 some point they need to learn how to deal with this.