I remember being told in high school “in a vacuum two objects will fall at the same rate no matter what the difference in mass (i.e. a quarter and a feather ).” The error in this phrase is perspective. A co-worker brought this argument to my attention:

Every object has gravity, though physics was so long ago for me I don’t remember the equation for calculating any objects gravity. So when you drop a quarter on Earth, the quarter will be “attracted” to the Earth since the gravity of Earth is much much greater than the gravity of the quarter. The same will happen with the feather. Using a vacuum only removes the air resistance. By the forementioned perspective I mean that if you were to drop an extremely dense object ( i.e. baseball sized cluster of neutrons) and say a bowling ball they might LOOK like they fall at the same rate from where you stand, but according to the math the cluster 'o neutrons are attracted to the Earth much quicker than the bowling ball.

the question: Why do they teach a flawed statement about physics?

The cluster’s attractive force will be much greater than that of the bowling ball. But that’s offset because of the fact the cluster is more massive, therefore takes more force to accelerate at the same rate.

I think what Orbytal is getting at is that the gravitational force between two objects is proportional to the product of their masses. So, the force would truly be greater for a falling quarter than for a falling feather.

Regardless, making it explicit in the “quarter vs. feather” experiment would only serve to confuse precisely those people you’re trying to clear things up for.

This statement: your OP
“in a vacuum two objects will fall at the same rate no matter what the difference in mass”

vs. this statement: your subject
"Why two objects do not fall at the same rate… "

to me are confusing…

Perhaps I missed something? Of course an object with more mass requires more force. But, where is it implied otherwise in the OP?

Again, I’m not trying to be an ass here. I’ve had a number of physics courses. So, I’m not completely ignorant on the subject.
Please set me straight in case I have misunderstood your point.

Perhaps the class was intro. and just did not get into details?

As I recall the gravitational force between to objects does relate to the masses of both objects. However, the mass of the Earth is so utterly overwhelmingly greater than the mass of a quarter or feather or bowling ball that the effect of their masses on the attraction is infinitesimal. We may be able to calculate it, but I don’t believe we can even hope to measure it.

There are several misconceptions and/or terminology issues in the OP:

Objects don’t have “gravity.” Objects have mass. All massive objects have an attractive force for all other massive objects. This force is called gravity. The magnitude of the gravitational force between any two objects is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them.

The force of gravity on the quarter is exactly the same as the force of gravity on the Earth in this situation. The force of gravity on the quarter causes it to accelerate toward the Earth. The force of gravity on the Earth causes it to accelerate toward the quarter. The difference is the magnitude of these accelerations. Since the Earth is so much more massive than the quarter, its acceleration is infinitesimally small. The acceleration of the quarter is much greater.

I’m not sure I understand where the rest of the OP is going, so I’m going to quit here.

Leaving general relativity aside, the feather and the neutron ball will fall at the same rate in a vacuum. Or more precisely, the earth and the feather will fall together at the same speed as the earth and neutron ball. Ultimately, it boils down to the experimental fact–that has no theoretical explanation I am aware of–that inertial mass and gravitational mass coincide, so the more massive object is equally harder to get moving. The two effects cancel precisely and you see things fall at the same speed.

As for general relativity, that does change things slightly, but objects have to get extremely massive and close together for that effect to be discernible.

Now why isn’t a straight line the shortest distance between two points. A straight line is DEFINED to be the shortest distance. If there are two geometries on the same points, then what is straight in one geometry may not be straight in another. So a straight line in the usual geometry of the surface of the earth is a great circle (or would be if the earth were perfectly spherical), while in the geometry of space, straight lines go through the earth.

Well, actually, if you dropped both items at the SAME TIME, it wouldn’t matter what the relative masses are since the earth will move up as the sum of their forces.

However, if you dropped one and timed it and dropped the other and timed it, you would have the more massive object hit slightly before the less massive one. The effect is so small, however, its not worth noticing.

F = GMm/R^2 ----- 1 (M is the mass of the earth, m is of the falling object)

Weight of the object (same as F above)

F = mg ----- 2 (g is the acceleration due to gravity)

Equating, 1 and 2, you’ll see that m cancels out and you are left with:

g = GM/R^2. It does’nt matter what m was.

Although, we see falling as the mass moving towards earth while the earth remains stationary or accelerates (This is important because strictly earth is not a inertial frame of reference because it has an acceleration - we neglect it because its so small) in reality both bodies accelerate, the earth does it very very minutely though.

Just as an historical note, this type of question has been discussed before. In searching for what I was remembering, I found “Does mass effect the rate of falling?”, Chronos summarizes the discussion in two previous, “hammer and feather” threads. Rather than post my own summary, I’ll copy his, since he already did it:

For more information, I highly recommend perusing the classud threads that Chronos links to.

There are many things taught in beginning physics that are not true in general.

For example, Boyles Law for gasses is only true for a restricted range of temperatures and pressures. Newton’s mechanics is only approximately true and fails at velocities approaching that of light in a vacuum.

So why pick on this one which is really a trivial exception?

The undergraduate courses are for the purpose of getting some general principles, tools to use them and to learn the terminology of the subject. When you go on to advanced study you find that what you learned as an undergraduate is only a first approximation.

There is one instance where what is taught is grossly wrong and that is the principle of least action. I was taught and probably most of you were that light follows the shortest path. Well that can be easily refuted by a sufficiently concave mirror in which light will follow (or appear to follow, which is the real issue) the LONGEST path. What it actually follows is any path along which the derivative of the distance is 0.

In fact, the light follows all paths. However, any path along which the derivative does not vanish is very very close to a path that is 1/2 wavelength longer and causes destructive interference and you see nothing. This effect can actually be verified by using a grating ruled in such a way that at the ambient wavelength there is a groove where the path is 1/2 wavelength longer and shorter, so you prevent this destructive interference. Then you can reflection at any angle.

But ignoring such gratings, light does not follow shortest paths; it follows extreme paths. With an elliptical mirror, it follows shortest paths; with a parabolic mirror, they are all the same length, but with a hyperbolic mirror, it is longest paths.

its only not true in some extreme examples and involving extreme gravity.

light shoots out of a star really far away, between the star and you is a black hole, the light, in order to reach you has to curve around/get bent by the gravitational pull of the black hole. even if the black hole is not directly between you and the star (as long as its gravity is strong enough to bend the light on its way past.
sorry if thats not crystal clear, I am ready for a nap here.

Doh!
I forgot one little detail, information cannot travel faster than light, therefore the path that light would travel between two points is the shortest distance between them.

These paragraphs are slightly contradictory. I believe the second one is correct. It seems to me that if they were dropped at the same time, the earth would indeed move up towards the dropped objects, but the part of the earth near the more massive object would be moving up faster than the part of the earth under the less massive object. Thus the more massive object would make contact with the earth first.

(Of course I agree that these differences will be too small to measure.)

Keeve, Shamalese is correct because there is a cumulative effect in the first paragraph (assuming the masses are adjacent).

If you went to opposite sides of the Earth and did this experiment simultaneously, the effect would be that the more massive object would make contact with the Earth twice the differential amount Shamalese mentions. These are all miniscule effects that can be seen to be due to Newton’s Third Law.

The vacuum is important pedagogically because it is what misled Aristotle (who believed that heavier objects fell faster than lighter objects). Removing air resistance does have the effect of allowing us to see the effects of few forces on the object we’re testing.

The general rule is that for changes in radial distances that are much smaller than the radius itself and for sufficiently sized masses there is a gravitational potential field that uniformly accelerates all objects. This is a fundamental result derived by Newton for explaining Galilean motion on the Earth and orbital motion of the planets.