x(m,t) attempt 1

Ultimately, it would be nice to have the position of a sheet be a function of the mass and the time. Potentially, this might mean we could use constant time-stepping rather than timestepping determined by the next sheet crossing (which slows the simulation down alot).

My first idea was to solve for the mass between 0 and position x at time t. Then I could potentially invert the relation to get x(m,t). Then we can take a time derivative to get the velocity as a function v(m,t). Here was my initial calculation:





This calculation can be seen in more detail by clicking on it, and zooming in with the browser.


This approach is failing because the sheet positions and ranks are still required, I had hoped they would drop out. In a sense at each position X all you need to know is the number of sheets interior to X, not their positions... but in order to know this at all X, you ultimately need the positions of all the sheets.

One question for Scott: in the x(m,t) picture you envisioned, were you thinking of hanging on to the discreteness of the particles, or were you thinking of a mass continuum?