Galileo theorized that in the absence of air.......?

Question

Galileo theorized that in the absence of air, all things would truly fall with the same acceleration. 300 years later the crew of Apollo-15 demonstrated this on the Moon (which has gravity but lacks air) by dropping a hammer and a feather. http://regentsprep.org/Regents/physics/p...

Question1

Suppose there are two sphere of diameters 1meter and 1 kilometer somewhere in the space at the same height from the surface of earth. The height is less than the distance between earth and moon and gravitational force of earth never goes away completely. So

Would they truly fall towards earth with same acceleration in space (absence of air resistance) down to layer where earth’s atmosphere start and then will change their accelerations. Now also apply to each mass F=GMm/r^2 . Would result be same?

Question2

Now if we dropped aforementioned two masses from the same height say 1.5 km above the surface of moon Would they fall at the same acceleration on the surface of moon, or

result would be different if we apply F=GM(moon)*m ( mass of 1 Km)/R1^2 , F=GM(moon)*m ( mass of 1m)/R2^2

Where R1 is distance between center of moon and center of mass whose diameter is 1 km and

R2 is distance between center of moon and center of mass whose diameter is 1 m

Please also note:

If they dropped on the surface of moon at the same rate by removing a wooden plank (say) from their bottom. Then application of F= GMm/r^2 to each mass will show different results.

Answer 1

It is true that two objects of different mass will fall at the same rate in the absence of air resistance.  You mention the equation F = GMm/r^2, but this can be simplified to (m*a) = GMm/r^2, or:

a = GM/r^2 .  As you can see, the mass of the object is completely taken out of the equation, so the acceleration of the falling object is constant regardless of the falling mass.

Of course, once the atmosphere is reached, the objects (assuming they are of equal size and shape, but unequal weight) will slow down due to the drag force against it.  The drag forces against each are equal, but the heavier mass will fall more quickly because the force exerted on it corresponds to a low (negative) acceleration:   a = .5*rho*S*U^2*Cd / m

Answer 2

1) Yes.

Force = Mass x Acceleration ..

If you want the acceleration of mass m, then divide the Force by m.



This will show that the acceleration of two objects (own = m & 'target' = M) depends only on the distance and mass of the 'target' object (M), not on it's own mass (m).

NB. The Moon and Earth are both spinning and moving (rotating about a common center) and the Sun and other planets (especially Jupiter) also impose a gravitational force on any objects in orbit.

Two objects will only undergo the same acceleration (due to  the gravitational force of a 3rd object) if they are equidistant from that object ...

As soon as the 3rd object is moving, then, since M & m start in slightly different positions, the distance from each to the 3rd can not remain identical (if for no other reason than because the 3rd object will be accelerated towards each of the first two at two different rates, one depending on m, the other on M))... and so the acceleration on m & M due to the 3rd object will change ...

Look up 3 Body Problem in wikipedia :-)