# Equation of motion with variable acceleration

1. Oct 8, 2006

### mitochondria

I have recently been working on a project regarding black holes and the spaghettification aspect of it interests me quite a bit. So, I have set out to try to derive some mathematical descriptions of the geometry of the object being spaghettify.

I have spent a few hours (uncessfully) trying to get an expression of acceleration in terms of time, which will eventually (hoepfully) lead me to an equation that desribes the changing distance (being stretched by differential force) of 2 furtherst points in the axis perpendicular to the gravitational field of the object (spherical) being spaghettified.

My problem is that I don't know the equations of motion in which the acceleration is not constant *frown*.

This is what I have so far for the two points described above:

$$F_{diff} =\frac{4GMmr}{x^3}$$

(for x >> r, where x is the distance between the two center of masses and r is the radius of the sphere)

Divide both sides by m:
$$a =\frac{2GMr}{d^3}$$

(I think...) If I want to get an expression of how r changes over time I need to integrate the expression with respect to t. In this case I need to find an expression for d in terms of t - which I can't because I don't know the equation of motions with variable acceleration...