A classical physics kinematic question. Time for a separation to reach infinity?

Question

Two masses, M1 and M2, are initially separated by a distance, d. A force exists between them that is proportional to the product of the masses and DIRECTLY proportional to the square of their separation, with a proportionality constant C.

F = C (M1) (M2) d^2

How long does it take for the separation to become infinite? Classical physics only, please.

This isn't a homework problem. I'm not taking any physics classes at the moment. I'm just curious.

The force is a repulsive central force, without curl, directly proportional to the square of the separation of the masses. I think the separation does go to infinity in finite time, assuming the rules of classical mechanics.

Let the masses be m and n, just to be neat.
F = C mn d^2

The force is related to the acceleration as
F = (m+n)a

The acceleration, really, is what counts, so
a = C r^2 mn/(m+n)

Define a new constant:
K = Cmn/(m+n)

Then, a = K r^2

da/dr = 2 K r
dv/dt = a = K r^2

We're trying to find r(t). But it looks like it's going to be a while before we get there.

r = (da/dr)/(2K)
r = [(dv/dt)/K]^(1/2)
a = (da/dr)^2/(4K)

(da/dr)^2 - 4Ka = 0

So, how do you solve a nonlinear first order diffeq of the form:

(y')^2 + Qy = 0

Answer 1

In Principal I guess it's possible that t approaches some asymptotic value as separation distance goes to infinity. The only way I can think of to solve it exactly is to find an appropriate variable substitution that makes the equation linear, or the asymptotic behavior.

Here's a suggestion. Several Classical Mechanics, and Orbital Mechanics, books present the equation for time in the central force problem in terms of an arbitrary Potential. For your problem V(x) ~ x^3. This is probably a better starting point. You may end up with an Integral that has a well defined asymptotic behavior.



What kind of Physics class is this? High School, College, Graduate school? Is it a General Physics, Classical Mechanics?

Are you sure the problem says the equation describes a Force? (Not Potential Energy?) If it is Force, any reference to direction?

I guess this depends on what you want to define as "effective" infinity.

Anyway, as stated and assuming the two masses are repelling each other, here's my shot, with the info. given we can define z = x1 - x2 as the separation distance between the two masses. This leads to the differential equation,

d2z / dt2 = C*(m1 + m2)*z^2

d2z / dt2 - second derivative.

This is a non-linear 2nd ordered equation. My guess is to treat the coeff. of z^2 like a time-constant, hence,

tau = power( CM, -1/2 ) - one-over-square root

I'm calling infinity that point at which a LINEAR exponential grows by a factor of 10, which is 2.3 times tau.

So, how about,

2.3 / sqrt( C*(m1+m2) )

Answer 2

Think about the problem before you answer, and you are less likely to be fooled by clever wording. The distance between them will become zero, not infinite. Unless of course the force is repulsive.

Answer 3

Never, you can't reach infinity