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Apparently, Aristotle reasoned (without experimentation) that heavier objects must fall to Earth faster than lighter objects. For example, a 1,000 kg ball of iron would fall faster than a 1 kg ball of iron.

Galileo is reputed to have disproved this experimentally by dropping two differently heavy objects from the Tower of Pisa, and they were observed to hit the ground at the same time. Case closed, right?

Well, it suddenly occurred to me this morning that Aristotle was actually correct, and Galileo's experiment was incapable of discerning any difference in arrival times between the two weights he dropped. And I wonder if our measuring devices today could discern the difference in arrival times in this case?

The formula for force due to gravitational acceleration is:

$$F = \frac{Gm_1m_2}{d^2}$$

For both the 1000kg and 1kg masses, the product of them both with the Earth's mass is clearly the Earth's mass, so $F$ will be virtually the same for them both. But if we take the 1000kg mass out to the Moon's orbit, and we stop the moon in its orbit so it stands still with respect to the Earth, the force $F$ will clearly be much larger between the Moon and the Earth than between the 1000kg mass and the Earth.

The result of this is that the Moon and the Earth will meet much faster than the 1000kg mass will meet the Earth.

Am I right, and is Aristotle vindicated, or is there something I'm missing here?

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    $\begingroup$ You missed the fact that the moon has MUCH more mass than the ball of iron. As acceleration = force * mass, the acceleration will be the same for both. $\endgroup$ – hdhondt Nov 27 '14 at 9:44
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    $\begingroup$ @hdhondt How can I miss the fact that the moon has MUCH more mass than the ball of iron? You think I can't tell that the moon has more mass than the ball of iron? How stupid do you think I am? $\endgroup$ – Cyberherbalist Nov 27 '14 at 10:00
  • $\begingroup$ In that case, if you apply a= f/m then the acceleration will be the same for both $\endgroup$ – hdhondt Nov 28 '14 at 4:18
  • $\begingroup$ Note that despite this being correct, Aristotle is not vindicated since Aristotle's claim was not "a 1000kg iron ball falls some microscopic fraction faster than a 1kg iron ball". His claim was that a falling object rapidly reaches a velocity proportional to its mass. This simply isn't true: even including air resistance so that there's such a thing as terminal velocity, the terminal velocity of similar iron balls isn't proportional to their mass. $\endgroup$ – Steve Jessop Jan 20 '15 at 12:54
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Consider three bodies $Earth$, $B$ and $C$. Now, we take two cases. In the first case, we will observe gravitational attraction between body Earth and the heavier body ($B$); and in the second case, we will observe the gravitational attraction between body Earth and the lighter body ($C$).

Case 1

Consider body $Earth$ and $B$, separated by a distance of $r$. Therefore, $$F_G=\frac{Gm_{Earth}m_B}{r^2}$$

Acceleration of $B$ is : $$a_B=\frac{Gm_{Earth}}{r^2}$$

And acceleration of $Earth$ is : $$a_{Earth}=\frac{Gm_B}{r^2}$$

Case 2

Consider $Earth$ and body $C$, separated by a distance of $r$. Therefore, $$F_G=\frac{Gm_{Earth}m_C}{r^2}$$

Acceleration of $C$ is : $$a_C=\frac{Gm_{Earth}}{r^2}$$

And acceleration of $Earth$ is : $$a_{Earth}=\frac{Gm_C}{r^2}$$

We see that $a_C=a_B$. But $a_{Earth}$ is changing when the mass is changing.

$$a_{Earth}=\frac{Gm_B}{r^2} ~~~\text{and}~~~ a_{Earth}=\frac{Gm_C}{r^2}$$

Therefore, $a_{Earth}\propto m_{body}$ and $a_{Earth}= \frac{Gm_Cm_{body}}{r^2}$

The acceleration of Earth is directly proportional to the mass of the body. So as the mass of the object increases, the acceleration of Earth increases, decreasing the time of impact and hence, decreasing the time taken by the body to fall.


Note - Although Aristotle was partially correct, he was far from the real understanding of Gravity. Galilean Gravity was closer to the truth. Aristotle deduced that heavy objects fall faster because his observation was affected by air resistance, which is not the real cause of heavier objects falling faster. Galileo had already refuted his explanation of gravity, explaining the affect of air resistance on falling bodies. Galilean Gravity was correcter.

Please refer to this answer for a detailed explanation.

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  • $\begingroup$ So, you're saying I am correct? $\endgroup$ – Cyberherbalist Nov 27 '14 at 10:22
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    $\begingroup$ Aristotle's explanation wasn't correct. $\endgroup$ – user49111 Nov 27 '14 at 10:33
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Clearly missed the point in this statement.

For both the 1000kg and 1kg masses, the product of them both with the earth's mass is clearly the earth's mass, so F will be virtually the same for them both.

The product is definitely not the same and the force on 1000kg ball is exactly 1000 times greater that the force on 1kg ball and much much more on the moon. But the acceleration being ${\frac{F}{m}}$ is going to be same for all these objects.

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  • $\begingroup$ Are you saying that the m in F/m is the same in both cases? That moon+earth = ironball+earth ? $\endgroup$ – Cyberherbalist Nov 27 '14 at 10:02
  • $\begingroup$ @Cyberherbalist No, I am saying the force is definitely higher for a massive body, but the acceleration (F/m) will be same because for a massive body both F and m are higher, but for a light body both are lesser. Or if we calculate acceleration from original equation F=G(m1m2)/d^2 and a=F/m2=G(m1)/d^2. The acceleration will always be same irrespective of the mass m2. $\endgroup$ – Sreekumar R Nov 27 '14 at 10:13
  • $\begingroup$ The question that is claimed my question is a duplicate of (but it isn't) actually has an answer that says I am correct when I say that the 1kg and 1000kg balls actually fall at different accelerations, but the difference is indiscerable. Compared to the moon and the 1000kg ball, however, there appears to be a big difference in acceleration. $\endgroup$ – Cyberherbalist Nov 27 '14 at 10:40
  • $\begingroup$ @Cyberherbalist This difference is immeasurably small. As for your question, Aristotle's reasoning of heavier bodies reaching first was completely inaccurate. I doubt if there was any reasoning at all. His Gravitational Law was completely empirical, without any mathematical backup, unlike that of Newton's or Galileo's $\endgroup$ – user49111 Nov 27 '14 at 13:34
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If you assume that the earth is immobile, the fall time if the same. If you consider that the earth moves because of the mass of the item, yes, the heavy item fall time is shorter (infinitesimal difference). If both items are dropped at the same time, they will hit the floor at the same time though.

(This question has already be answered : Don't heavier objects actually fall faster because they exert their own gravity?)

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  • $\begingroup$ Actually, my question is about Aristotle actually being correct (so far as I understood what he claimed), and not specifically about heavier objects falling faster -- although that does enter into it. @ThePragmatick addresses Aristotles reasoning, which had to do with air resistance, not mass, so my question had an incorrect basis. But it's still not a duplicate to the question you reference -- although that question yielded some excellent answers, and thanks for pointing it out! $\endgroup$ – Cyberherbalist Nov 27 '14 at 10:38

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