Re: [Phys-L] Bernoulli's equation

[John Denker <jsd@av8n.com>]

On 01/25/2013 01:54 PM, Anthony Lapinski wrote:
> Thanks to all who responded. This is for my (high school) honors physics
> class (non calculus).
>
> Of all the equations we derive (range equation, work-energy theorem,
> nonconservative work, impulse-momentum theorem, etc.), this one is the
> toughest. So that's why I wanted something "simple."


Suggestion:  You can argue that pressure is plausibly related to
ρ v^2 on dimensional grounds, and then stick in the factor of ½
by fiat.  The sign of the effect is simple to understand:  when
a parcel is accelerating, it must have higher pressure behind
and lower pressure ahead.  That gives us
  P + ½ ρ v^2 = const                          [1]

We "could" derive the factor of ½, but we are not going to, not 
at the moment anyway.  The simple derivations are not correct, 
and the correct derivations are not simple.

Note that equation [1] is a special case of a more-general
equation.


Leaving the derivation for later is a disappointment but not a 
disaster.  There are other examples of that ilk, which don't 
seem to bother anybody.  For example, consider the formula for 
the kinetic energy of a low-speed particle: 
   KE = ½ m v^2 
      = ½ p • v       (better)                 [3]

You can argue that KE is plausibly proportional to p • v on
dimensional grounds, and then stick in the factor of ½ by fiat.
We "could" derive the factor of ½, but we are not going to,
not at the moment anyway.  This factor is utterly nontrivial,
as you can see from the fact that for photons,
  KE = p • v       (with no factor of ½)       [4]

Note that equation [3] and equation [4] are special cases of a
single more-general equation.  Deriving these equations is one 
of the niftiest calculations in all of physics, provided the 
students know anything about the geometry and trigonometry of 
spacetime, which alas they probably don't.  The derivation can 
be found at:
  http://www.av8n.com/physics/spacetime-welcome.htm#eq-e-mc2-small-p2


I don't have any problem saying we will derive these factors of
½ later.  It gives the students something to look forward to.  
Similarly, mentioning that equation [1] is an expression for the
enthalpy (not energy) per unit volume gives them a reason to pay 
attention when enthalpy is eventually introduced.