The Straight Dope answer located at this URL: http://www.straightdope.com/mailbag/mgravity.html is incorrect. It seems that the supplier of the answer did not take the time to consider the question throughly or did not even read the question without jumping to the conclusion that they already had the answer.
The reason is simple. All Ian managed to do was prove that the acceleration of the hammer toward the Earth will be the exact same as the acceleration of the feather toward the Earth. However, what Ian, and most of the “scientists” to whom the writer who asked the question referred, failed to consider is the acceleration of the Earth toward the object. It’s clear that not only is the Earth attracting the objects according to the Universal Gravitation Constant, but the objects themselves are also attracting the Earth. This attraction, of the Earth toward the objects, is what differs. The hammer attracts the Earth at a higher rate of acceleration than the feather attracts the Earth.
This acceleration, while incredibly miniscule, does certainly exist. It can be determined by dividing the Gravitational Force between the Earth and hammer by the mass of the Earth instead of the mass of the hammer. And it will cause the hammer to make contact with the Earth before the feather, regardless of how imperceptible the difference may be.
Howdy, Huck. Welcome to the SDMB. Pleased to have another thinker about physics questions on the board. Hang around a while and explore. You’ll have fun.
I must warn you, however, that extreme caution is advised for any math-phobic person entering these threads. I will not be responsible for any exploding heads they may suffer. (Also, with the recent board software change, some of the formating is messed up, so they are even harder to read than they otherwise would be.)
So, Huck, you may well be getting more than you wanted to know, but enjoy.
Galileo, a Hammer, & a Feather is my favorite thread! Opus1 hit upon the right answer. It’s great because it’s a pithy little thing to remember. If anyone ever brings it up, just say (in a snooty voice) “I can’t remember all the calculations, but everyone knows the Heisenberg uncertainty swamps any difference due to the Earth’s acceleration.” Then roll your eyes. The detour through evaporating black holes just added to the whole effect…
You know, reading the OP was deja vu–I even went searching through the old threads, convinced that I’d find it somewhere. Clearly, though, Huckleberry needs to make the assumptions clear:
[ol]
[li]Are the hammer and feather dropped at the same time?[/li][li]If not, is the hammer on the ground when the feather is dropped?[/li][li]Are the hammer and feather point masses or are they a real hammer and feather?[/li][li]Is the hypothetical earth spherical (except for the possibly real hammer and feather, perhaps)?[/li][/ol]
The feather falling alone will hit the earth in some time t that depends only on the height. The feather accelerates toward the earth at 9.8ms^-2; the earth’s acceleration is negligible.
If the mass of the hammer is equal to the mass of the earth (as someone suggested in one of the previous threads), the hammer falling (without the feather) will hit the earth in time (t divided by the square root of two), or about 0.7t. The time here is less because the earth and hammer will meet not at the original surface of the earth, but at the center of gravity, halfway between the original positions of the hammer and the earth. Both the hammer and the earth will accelerate at 9.8 ms^-2.
However, if the feather and hammer are dropped together (simplification: they are touching each other when they are let go), the earth will accelerate toward both the feather and hammer, and they will reach the earth at the same time, at 0.7t.
These results can be generalized to show a relatively massive object dropped alone will reach the surface of the earth in a shorter period of time than a relatively light object dropped alone. But if both are dropped at the same time, they will reach the earth at the same time.
WillGolfForFood summed up the misunderstanding perfectly, way back in March.
Someone in that link that ZenBeam posted calculations that showed that they reach the earth in the same amount of time, in both cases. The key is that the hammer is lying on the ground when you drop the feather–thus adding just the right amount of mass to make the times equal.
But I’d be sceptical of such calculations, if I were you.
Well, it seems to me that all the calculations and thoughts thus far have forgot one critical element: TIME itself slows down with velocity. Hence, the velocity of the earth moving toward the hammer slows down time (relative to the viewpoint of a traveller on the hammer, say a little bug with a stopwatch.) Now, once you’ve got a bug with a stopwatch on the hammer, the whole question of the mass of the hammer becomes irreverent, since the bug (not necessarily being Catholic) wouldn’t celebrate mass at all. However, the mass of the hammer also decreases with it’s velocity. Thus, at some point in the trajectory, the mass of the hammer will exactly equal the mass of the feather.
I further note that the hammer can never hit the earth at all, because there are two Greek ducks observing the experiment… and you recall Xeno’s para-ducks.
The detour through evaporating black holes just added to the whole effect…