The Time That 2 Masses Will Collide Due To Newtonian Gravity?

Question

My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r. Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as: $$F_g ~=~ G\\frac{Mm}{r^2}$$ How do we find out at what time they will collide? What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$. I've tried taking a number of integrals, which haven't really yielded anything useful. Any guidance? No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :)

Answer 1

This problem is easily solved by applying principles of classical Newtonian Mechanics.

The force exerted by M on m is given by GXmXM/(r^2) N = ma

Therefore acceleration a of m towards M = GM/r^2 which is variable, function of r.

According to classical mechanics if a is constant distance s = ut + (1/2) a t ^2 where u is initial speed = 0

because masses are at rest. The distance s is separation between masses which is r. Therefore above equation for our scenario is r = (1/2) t^2 GM ∫ (1/r^2) dr = (1/2) t^2×GM/r. Integration limits r to 0.

We can solve for t = r [2/GM]^1/2 sec. Please let me know how do you like this answer by sending me E-mail on WYZANT message.