Re: [Phys-l] ballistic arc length

[John Denker <jsd@av8n.com>]

On 12/28/2009 04:20 PM, Peter Schoch wrote:

> A projectile is launched with an initial speed v_0 at an angle  
> _theta_. 
> .... we want to find something about the actual path of flight:
>
> 1.  Find the equation for the path of the trajectory (arc or the  
> flight path).

Easy.

> 2.  Find the angle that would minimize/maximize this path.
>
> Now, using basic rules of calculus I can find the first answer of an  
> arc length (which is a very complex experession).  Differentiating  
> this to find 2 is proving daunting.
 
The physics is easy.

The math is completely routine and straightforward, 
but ugly.

> 1.  Is there an easier way to approach this? (Hamiltonian with  
> appropriate coordinate system?)

I doubt it.

As the saying goes, if a complicated procedure yields a
simple result, you should look for a simpler procedure.

OTOH if the answer is known to be ugly, there's little
hope of finding it via a simpler procedure.

The arc length as a function of theta is plotted here:
  http://www.av8n.com/physics/img48/cannon-arc.png

The max arc length occurs when theta is the solution to:
   tan(theta) = sinh(1/sin(theta))

i.e. when theta is numerically rather close to 1 radian.
In more detail: 0.985515(1) radians, as you can
see from this plot:
  http://www.av8n.com/physics/img48/cannon-arc-root.png

I'm way too lazy to do the math myself, when I can get
a computer to do it for me.  The maxima (macsyma) script
is here:
  http://www.av8n.com/physics/cannon-arc.max

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

Since the problem can be stated in simple terms, one
might have hoped that the solution would be simple ...
but sometimes it doesn't work out that way.

(The four-color map theorem can be stated in simple
terms, but the proof is not simple.)