Speed, acceleration, accelerating acceleration, etc. How do we know where to stop?

Question

I am not a physicist.

Suppose a body A is falling towards body B in a vacuum. We know that A's speed will increase. However, as A draws near to B, the force of gravity will increase so the rate at which A accelerates will increase. Also we can presume that B's motion is affected by A. So now we have multiple levels of acceleration.

Question

I understand that, for practical purposes we can usually neglect smaller effects. My question is: In Nature do we ever get to the end of this apparently bottomless pit of derivatives?

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Considerations

A and B are of comparable but non-identical mass. They will therefore accelerate towards one another at differing rates.

When they get close enough, they can no longer be assumed to be a dimensionless point wrt gravity.

Answer 1

In your question, you are tracking both ${\bf r}_A$ and ${\bf r}_B$ (the position vectors in a fixed coordinate system), with something like:

$$ m_a\ddot {\bf r}_A = km_am_b\frac{{\bf r}_A - {\bf r}_B}{||{\bf r}_A - {\bf r}_B||^3}$$

$$ m_b\ddot {\bf r}_B = km_am_b\frac{{\bf r}_B - {\bf r}_A}{||{\bf r}_A - {\bf r}_B||^3}$$

which is not an endless pit of derivates, but rather a endless cycle of second derivatives.

However, there is a coordinate transformation that helps. If you rewrite the equations in terms of:

$$ {\bf R} = \frac{m_a{\bf r}_A + m_b{\bf r}_B}{m_a+m_b}$$

$$ {\bf r} = {\bf r}_A - {\bf r}_B $$

you should find:

$$ \ddot {\bf R} = 0 $$

and

$$ \frac{m_am_b}{m_a + m_b}\ddot{\bf r} = km_am_b\frac{{\bf r}}{||r||^3}$$

which breaks the cycle.

The first coordinate is the evolution of the center-of-mass: since there are no external forces, it moves at constant velocity.

The other coordinate is just the separation, which works when you use the reduced mass ($\mu$), defined via:

$$ \frac 1 {\mu} = \frac 1 {m_a} + \frac 1 {m_b} $$