Acceleration of gravity in space.

[loafula]

I was wondering if someone could answer a question for me. If you had two objects in space X distance from each other, is there a formula to determine what their velocity would be at the point where gravity finally pulls them together? Let's assume that the objects smack into each other instead of orbiting each other, and that there are no other forces acting on the objects.

 

[tiny-tim]

Hi loafula!

Yes, it's conservation of energy …

KE + PE is constant, so if you know the original PE ( the potential energy ), you can find the KE, and therefore the speeds.

 

[loafula]

Thanks Tim!
I understand that KE would equal the PE from the objects initially moving apart, but is there a way to determine velocity two objects would have at any given point along their paths toward each other? Assume the two objects are of equal mass, and have an initial velocity of zero.

 

[tiny-tim]

Equal mass or not, it'll always be minus the PE of their relative position.

 

[Janus]

Thanks Tim!
I understand that KE would equal the PE from the objects initially moving apart, but is there a way to determine velocity two objects would have at any given point along their paths toward each other? Assume the two objects are of equal mass, and have an initial velocity of zero.

Find the PE of the objects at their initial distance.
Find the PE of the objects for the distance between them at the given point.
Take the difference.
Use this answer and the formula for kinetic energy to find what velocity either mass has at that point.

 

[fawk3s]

cant you just find the acceleration with the gravity's formula and Newton's second law's formula? (if I am worng, please don't kill me). wouldn't be any point to it though after what you guys said.

 

[Integral]

cant you just find the acceleration with the gravity's formula and Newton's second law's formula? (if I am worng, please don't kill me). wouldn't be any point to it though after what you guys said.

I am understanding that we need only consider the forces between the 2 bodies.

We have
$$F_{12} = F_{21}= F = G \frac {m_1 m_2} d$$

Where d is the distance between the 2 bodies.
We also need for d to be much greater then any dimension of either of the bodies.

For the acceleration of body 1 we have:
$$a_1 = \frac F {m_1}$$

For body 2
$$a_2 = \frac F {m_2}$$

Now set up a coordinate system with the Origin at the Center of Mass of the 2 bodies.

You now have 2 differential equations from which you can find the speed and location of either body and any time.