# [Maxima] Shouryya Ray's equation

Jaime Villate villate at fe.up.pt
Fri Jun 1 09:39:16 CDT 2012

```On 05/31/2012 05:22 PM, dlakelan wrote:
> This is fine as a toy math problem, but the proportionality in real
> physical problems is not a constant.
Of course, we are talking about a mathematical problem; namely, an
analytical method to obtain the solution
to a system of non-linear ODEs.

> In fact it's a function of the Reynold's number (and the shape). I
> used the spherical projectile problem with air drag in a class on
> programming in matlab where students solved it using numerical methods
> to find things like the range of a particular projectile, and answer
> questions like whether the mythbusters could put their cannon ball
> through your window given some initial conditions (a reference to an
> unfortunate accident that the producers of that TV show had, where
> thankfully by some miracle no one was hurt).
>
> The formula I used was taken from this paper (link below)
Agreed; in my lectures I also give the typical example of a spherical
projectile to illustrate the use of numerical
methods for systems of ODEs. Looking at one of the examples I usually
solve, a tennis ball with radius 3.25 cm
and mass of 62 g, thrown with an initial speed of 12 m/s, at an angle of
45°, and using the formula you suggest,
I can see that the drag coefficient varies between 0.400 and 0.407, so,
in my opinion, the approximation that
Cd is constant is very reasonable in this case. Of course, this example
and all the other examples I give to my students are also toys created
by theoretical physicists.

In a real situation I would be worried  about other very important
factors that I believe are not taken into account in the formula you
referred. Wind speed? ball rotation? Magnus effect?

I have not made any experimental measurements and I do not play golf,
but I would be very surprise if a real-life projectile had a perfectly
plane trajectory like the ones I model using numerical methods :)

Cheers,
Jaime

```