What is the velocity due to gravity between two point masses?

[cromata]

Homework Statement

-Two objects of negligable radius and masses M and m are fixed in space at the distance R. Gravity is the only force acting between those two bodies. At some point in time, body of mass m is realized and starts moving due to gravity toward statical mass M.
Find v(x) and speed of body m when it collides with M [x is the distance between the objects]

Homework Equations

F=GMm/r²

The Attempt at a Solution

GMm/(R-x)^2=m dv/dt
GM/(R-x)^2 =dv*v/dx
GM ∫dx/(R-x)^2=∫v*dv [integral of dx goes from 0 to x, and v from 0 to v]
GM*[1/(R-x)-1/R]=½(v^2)
v(x)=sqrt(2GM/R) * sqrt[x/(R-x)]
and for this function, we get that the speed at the moment of impact (x=R) is infinity
-Same result is obtained using energy conservation law, but it doesn't make sense to me

 

[kuruman]

GMm/(R-x)^2=m dv/dt

If x is the distance between the two objects as you claim, is this expression consistent with Newton's Law of gravitation?

 

[cromata]

I made mistake, I found v(R-x)... but I'm not interrsted in that part of the problem (it's trivial). I just want to know why is v=infinity when two objects collide. Is it because in this sort of the problem we can't say that radius of objects is negligable?
-Because this result doesn't make any sense, for example if we have some object (some asteroid or smth) falling on Earth and we say that Earth's radius is negligable in comparison with the starting distance between Earth and that object, we would obtain that speed of that object is infinite when it hits Earth.
-Also if we have 2 small charges that attract each other (one is fixed, other is free to move) and distance between them is much larger than their radius. We would again obtain same result: infinite speed at the moment of collision.

 

[kuruman]

The result that v = infinity when the two objects collide is an artifact of the model that assumes that there is such a thing as a point mass or point charge. Reason it out. You start with zero kinetic energy and finite potential energy. As the objects come closer, you convert potential energy to kinetic. Since the initial kinetic energy is zero, the instantaneous kinetic energy at point x is the absolute value of the change in potential energy. What happens to that change at x = 0 where the potential energy is infinite according to the model?

 

[cromata]

Gravitational potential energy at the beging of the motion is finite number (Egp1), and when they collide Egp2=-infinity
Ek(kintetic energy at the moment of colision)=|Egp1-Egp2|=infinity, hencr v=infinity. So the problem is in using point masses

 

[kuruman]

So the problem is in using point masses

Yup.