The questioner claims that, since the overall mass of an earth-hammer system is greater than the overall mass of an earth-feather system, a hammer dropped from, say, the Tower of Pisa (in a perfect vacuum), will hit the ground sooner than a feather. The respondent claims this is false, and both will hit at the same time (see Galileo’s famous experiments not involving the tower). But the questioner is indeed correct: the hammer will cause the earth to accelerate toward the hammer more than the feather will, and the hammer will land ever so slightly sooner. Detailed explanation follows:
Suppose we have two masses: m1 and m2. Let m1 be the hammer/feather and m2 be the earth. The gravitational force on each is F = G m1 m2 / r^2. Now, m1 a1 = F, and m2 a2 = F, implying that a1 = G m2 / r^2 and a2 = G m1 / r^2. Assuming m1 << m2, a2 is, practically speaking, negligible, and a1, and hence time for m1 to hit m2, is independent of m1. This will be observed experimentally.
But, in reality a2 > 0 (the earth is not truly a fixed point), and therefore the overall acceleration of the two bodies towards each other is actually dependent upon both m1 and m2. For example, suppose m1 = m2 = m. Then the overall acceleration is roughly twice that of a hammer/feather falling to earth (i.e. two earths “falling” toward each other).
Anyhow, thought I would point this out, as it is technically true that heavier objects do, in fact, “fall” faster than lighter objects. Unless I’m greatly mistaken?
See that they all reach the ground at the same time.
Repeat the experiment, only this time, put two of the boxes right next to each other and drop them.
Note all three boxes hit the ground at the same time.
Repeat again, only this time, the two boxes that are next to each other will actually connect to form one box that is twice the mass as the remaining box. And drop them.
Note that the two boxes hit the ground at the same time despite one box being twice the size of the other.
Hopefully this thought exercise demonstrates why heavier objects don’t fall faster than lighter objects. The heavier objects are just lighter objects attached to each other. Attaching them to each other doesn’t make them fall faster.
I agree that in this thought experiment it does not matter whether two of the three boxes are fused to create a more massive box. But isn’t this because the earth will be accelerating toward the three-box system at the same rate in every scenario? If we had just one or two boxes, then the acceleration of the earth towards the one/two box system will vary, depending on the number of boxes.
This thought experiment also holds if the earth is a perfect fixed point. But my fundamental point is that, in reality, we cannot assume the earth is a fixed point.
Okay, revision: If a hammer and feather are dropped together then yes, they will collide with the earth at the same time. I think the questioner’s point, and mine, was that a hammer dropped by itself will reach the ground ever so slightly sooner than a feather dropped by itself would.
But if you’re dropping just the feather, where’s the hammer? On the ground, part of the Earth. And when you drop just the hammer, the feather must be on the ground, part of the Earth. So the Earth has more mass when you’re dropping the feather than when you’re dropping the hammer.
This got covered in great detail in another thread about 12 years ago, that I’m too lazy to search for. Bottom line, uncertainties due to the Heisenberg uncertainty principle swamp any difference you calculate using Newtonian physics, no matter what assumptions you make.
The flaw in the OP’s argument is that it only accounts for force, but acceleration is f/m.
In the case where the masses are greater, the force is greater but … so are the masses! Yes, the force between Earth and hammer is greater, by exactly the amount needed to accelerate them towards each other just as fast as the earth/feather system.
The key is that the mass that causes the gravitational force is identical to the mass that causes inertia. Maybe in some kind of universe they might not be equal, but in ours, they do seem to be, and Einstein’s general theory of relativity is based on the assumption that they are. This assumption has been verified to an astonishingly high precision.
Scratch my post above, which doesn’t address the OP’s claim. On further reflection, I agree with the OP’s claim, with the caveat expressed by ZenBeam.
However, the point about the heisenberg uncertainty principle won’t hold if we vary the masses considerably. At some point, we have a weight heavy enough to measure the difference, and the question becomes “How heavy a hammer do we need?”
I was about to disagree with you, then I saw your new post. I think ZenBeam nailed it, but this is an interesting twist to a problem that I thought I really understood until today.
If I’m calculating it correctly, the added mass to the Earth would not quite even it out. The hammer would still hit in ever so slightly less time.
When calculating it, I used a much larger difference in mass so I didn’t have to calculate it out to so many decimal places but there should still be a difference, just to a lesser degree.
Here’s a hammer/feather/vacuum race that was captured on film some years ago. Well worth checking out both to learn the results and because it was quite expensive to mount:
As long as we’re being ridiculously precise, I pose an entirely different answer to what happens if you drop the hammer in a vacuum, and then drop the feather at some later time. The answer is: We don’t know. It depends.
Yes, the hammer pulls the Earth towards it as it falls. But there are lots of other masses in motion. The tides are rising and falling. Rain and snow are falling. Maybe somebody on the opposite side of the Earth is dropping an identical hammer at the same time. Mars, Venus, and the Moon might have some effect.
If you’re timing the drop times precisely enough to detect the Earth rising to meet the hammer, then your measurements will be flooded by all sorts of other forces acting on the Earth. No matter how many times you do it, you’ll always get a slightly different answer. You can suck the air out of your experimental chamber, but you can’t keep out gravity. And gravity is always changing.
This is such a facetious point, it makes Zenbeam lose a lot of credibility’.
Because whether the feather or the hammer are part of the Earth when you’re dropping the other object, is completely beside the point, it’s irrelevant.
The original question is if heavier objects fall faster than lighter ones. If you think the hammer makes the Earth a larger mass, then rephrase the question in such a way that the other object is completely out of the picture. If you wanna get specific about it, imagine that it floats around Mars or where ever.
Now, to the original argument that because of the heavier mass the hammer will fall faster:
This such an infinitesimal small difference, it’s doubtful it can be measured.
An average hammer would be something like 500~1000 more massive than the average feather, so the F would be greater for the hammer. But, with that stupendously greater mass, the hammer would distort spacetime as much as a thousand times more than the feather would, creating a “huge” difference in time dilation WRT to general relativity. Hence, it is entirely possible that the hammer might fall faster in reality, but we could never know because we can only observe the event from a relative perspective, either that of the person dropping it, a spectator, or a passenger on one of the falling objects.
In the interest of maintaining credibility as a “good source of truth” I would think that the article should be updated. Particularly since the article reply is somewhat snide ( “Um … yeah, I’ll set someone straight, B, I promise.” ). The article is strictly incorrect because it says “In this case, we are looking for the acceleration of the falling object, m2”. No, we’re looking for the combined acceleration of both objects, ie, the falling* object and the target it is falling towards.
Babynous (the person posing the question) isn’t correct either: as has been pointed out in this thread, if both the heavy object the light object are moving towards the target at the same time, then they will hit at the same time because the target is accelerated towards them by the sum of their masses. But if treated individually, the heavier object will hit first precisely because of effect Babynous describes: it will accelerate the target towards itself more than the lighter object accelerates the target.
*Poor terminology since they’re both falling towards each other
