Originally Posted by

**TKHunny** Off the top of my head, I'd say you would have to ask the concept or Relativity whether or not the relative acceleration is constant. Personally, I'd be a bit shocked if it were constant.

You have $\displaystyle D_{x} \cdot D_{y} = r^{2}$

Since the position of Y is dependent on the position of X, we also have: $\displaystyle D_{y} = f(D_{x})$ (Which you might have stated in a less abbrevaited preamble.)

This leads nicely to the implicit derivatives:

$\displaystyle D_{x} \cdot D_{y}' + D_{y} = 0$

and

$\displaystyle D_{x} \cdot D_{y}" + D_{y}' + D_{y}' = y$

Then, just a little algebra produces:

$\displaystyle D_{y}' = -\frac{D_{y}}{D_{x}}$

and

$\displaystyle D_{y}'' = -\frac{2 \cdot D_{y}'}{D_{x}} = \frac{2 \cdot D_{y}}{\left( D_{x}\right)^{2}}$

Frankly, this is a bit astounding. As X gets close to the Origin, the crew on Y had better batten down the hatches! If X only flinches, Y will jolt off a very long way... One cannot argue much with that SQUARED Dx in the denomiator. Wow!

Remember $\displaystyle \Delta E = \Delta m \cdot c^{2}$ from the Special Theory of Relativity? It HAD to be astounding just how much energy differential would be created by a very small amount of converted matter. Same thing. Giant numbers in the numerator or tiny numbers in the denominator are bad enough. SQUARE them and forget it. That's BIG!