B Gravitational force between two objects?

[tummbacoco]

I know that the force of gravity is (ma)=GMm/r^2 or a=GM/r^2
This makes sense and If I were to drop a bowling ball down to Earth I'd expect it to fall with 9.8m/s^2. However I can calculate that the bowling ball has its own gravity using the formula noted above. My question is what will be the true acceleration of these 2 objects towards each other?

Perhaps a better example is the Earth and say Jupiter. Would these 2 planets go towards each other at a rate of 25m/s^2(gravity of Jupiter), or perhaps they would go towards each other with 35m/s^2(Gravity of Jupiter+Earth) because the they are both pulling each other in right?

So this question seems so simple that any info I've found on it, neglects an explanation. Anyways, thanks!

 

[Spinnor]

You know the force of the ball on the Earth and you know the Earth's mass. What is the acceleration of the Earth towards the ball? So they both move towards each other.

 

[tummbacoco]

You know the force of the ball on the Earth and you know the Earth's mass. What is the acceleration of the Earth towards the ball? So they both move towards each other.

So you are saying that adding up their accelerations would be the true acceleration of gravity? By that logic, does that mean that an object of higher mass would indeed fall faster than a lower mass object due to the fact that the higher mass object pulls the Earth towards it more, thus shortening the time it takes for it to reach the ground?

And this wouldn't even violate Newtons law because the Earth is pulling on both objects at the same acceleration, it's just the more mass the more that the Earth gets pulled towards the object?

 

[Nugatory]

So you are saying that adding up their accelerations would be the true acceleration of gravity? By that logic, does that mean that an object of higher mass would indeed fall faster than a lower mass object due to the fact that the higher mass object pulls the Earth towards it more, thus shortening the time it takes for it to reach the ground?

There is an important subtlety here. You're talking about the acceleration of the object... but is that the acceleration observed by someone standing on the surface and considering himself to be at rest? Or is it the acceleration observed by some distant observer watching the Earth and the falling object being pulled towards one another? These will not be the same, and there's no particular reason to call either one "the true acceleration of gravity".

 

[Khashishi]

And this wouldn't even violate Newtons law because the Earth is pulling on both objects at the same acceleration, it's just the more mass the more that the Earth gets pulled towards the object?

If you drop an object on Earth, the gravitational force of the Earth on the object is the same as the gravitational force of the object on the Earth. There is no violation of Newton's law (until you get into extremes requiring general relativity).

The two-body problem orbits are elliptical (unless they crash into each other). If you want to solve the trajectory motion, you can reduce the two-body problem to a one-body problem, using the concept of reduced mass.