Motion under a direct-square force.

[Jenab]

I've heard it said that two particles repelling each other by a force that increases in direct proportion to their separation will raise that separation to infinity in finite time...under classical, non-relativistic physics, of course.

Does anyone happen to have the derivation?

Jerry Abbott

 

[Tide]

Just integrate the equation of motion which can be written in generic form as

$$\frac {d^2x}{dt^2} = x^2$$

to find

$$\frac {dx}{dt} = \sqrt {v_0^2 + \frac {2}{3} \left( x^3 - x_0^3 \right)$$

from which it follows that the particles can make it to infinity in finite time.