Re: [Phys-L] time constants and exponential decay

[John Denker <jsd@av8n.com>]

On 06/15/2017 09:49 AM, Philip Keller wrote:
> I just learned an interesting geometric way to think about those time
> constants: from any starting point during the decay, the tangent line
> always takes the same amount of time to reach the asymptote.

This is an example of a *scaling law*.

If you scale the ordinate of an exponential by some factor, it's still
an exponential, with the same time constant.  The tangent (and its
zero-crossing) go along for the ride.

You don't need calculus (let alone differential equations) to prove
this.  A plot on semi-log paper suffices.

(You can also also get there via dimensional analysis.  As always,
dimensional analysis is a quick-and-dirty approximation to a some
sort of scaling argument.)

Scaling laws are a reeeeeally important part of physics, and have
been since Day One of modern science (1638).  They are remarkably
convenient, given how powerful they are -- and vice versa.

   https://www.av8n.com/physics/dimensional-analysis.htm
   https://www.av8n.com/physics/scaling.htm

A few days ago I suggested that one should never pass up the chance
to make a symmetry argument.  By the same token, one should never
pass up the chance to make a scaling argument.