Gravity Pull: Calculating Time to Impact

[O-r-i-o-n]

There are two objects with the distance of r apart .
because of the gravity , these two objects start to get closer to each other and then they get to each other (actually they fall on each other) .So my question is how long does it take for them to get to each other?
Thank you

 

[Astronuc]

Do the two objects start at rest?

Is this a homework problem?

 

[O-r-i-o-n]

Yes , their initial velocity is zero .
No it's not a homework , the teacher told us how to use Kepler's third law for a big fall like this (for example an asteroid falls to the sun) but in those kinds of cases one object is a lot heavier than the other one.
so this question popped in my head what if you can't ignore the small object's mass?
thanks

 

[Janus]

If you can't ignore the mass of one of the objects, then you need to use a modified form of the orbital period formula where you add the masses together (where you normally would have "M", you substitute (M+m). Then take half of the orbital period to get your fall time.

 

[O-r-i-o-n]

Good , I didn't think of that , thanks
but once someone solved this but I was too young to understand what he did
He was trying to solve the " m\ddot{r}=-gMm/r^2 " differential equation , but I guess your solution is good , too . Thanks