Thread: Time for two bodies to collide under Gravity

I am trying to compute the time it takes for two point masses to collid under gravity.

For example, consider two one kilogram masses 1 meter apart. Can someone give me a hint to compute the time it takes for the masses to collide.

I know how to find the velocity at which they will collide. That can be obtained by setting the initial gravitational potential energy to the kinetic energy of the two bodies. I could then solve for V.

Find out what the force is between the two objects.

From that work out the acceleration of one object.

Equations of motion... bobs yer uncle.

Originally Posted by sox

Find out what the force is between the two objects.

From that work out the acceleration of one object.

Equations of motion... bobs yer uncle.

There's a little bit more to it than that. The force between the two object's changes as they get closer, as does the acceleration.

On way to approach this problem is to treat each object as if it were at the periapis of an extremely eccentric orbit, then compute half the period of such an orbit.

For two objects in mutual orbit the period is:



Where a is the semimajor axis of the orbit and in this case would be would be 1/4 the distance between the centers of the objects.

Of course, this assumes the "collision" occurs when the centers of the objects meet.

Hi Janus

I read on Wikipedia

"Note that for all ellipses with a given semi-major axis the orbital period is the same, regardless of eccentricity."

That is important, otherwise your method will not work.

However, for the case where the two 1kg objects are 1 meter apart then I should used a semimajor axis of .5 meters right.

So the time would be ?

Originally Posted by ScubaDiver

Hi Janus



However, for the case where the two 1kg objects are 1 meter apart then I should used a semimajor axis of .5 meters right.

So the time would be ?

No, for two reasons:

The time from maximum separation to collision is 1/2 the period of a full orbit, so you need to drop the 2

The focus of each object's orbit is the barycenter of the system, which in this case is the halfway point between the two. Since we are treating this as an extremely eccentric elliptical orbit, this focus will be at one end of the major axis of the ellipse and the starting point of the mass will be at the other end. Thus the semimajor axis will be 1/2 of this or 1/4 the separation of the two masses.

Ok, I understand what you mean know. I drew out a few pictures and now it is clear.

Originally I was imagining the orbits incorrectly. If the orbits are extremely eccentric, then the objects will essentially come straight at eachother, and then they will get whipped around eachother rapidly and shoot away from eachother.

So anyways, plugging in some numbers for two 1kg masses separated by 1 meter I would get



which is approximately 10,822 seconds
which is approximately 3 hours

Originally Posted by ScubaDiver

Ok, I understand what you mean know. I drew out a few pictures and now it is clear.

Originally I was imagining the orbits incorrectly. If the orbits are extremely eccentric, then the objects will essentially come straight at eachother, and then they will get whipped around eachother rapidly and shoot away from eachother.

So anyways, plugging in some numbers for two 1kg masses separated by 1 meter I would get



which is approximately 10,822 seconds
which is approximately 3 hours

Oops, I think you forgot to multiply by pi.

Haha, yes I did

So the final answer should be about 9.44 hours