# Is Galilei's reasoning on free fall valid?

Galileo Galilei discovered by experiments that all bodies tend to fall with the same rapidity (I use it in an intuitive sense, you can replace it by 'acceleration' used in today's physics language), independently of their weight. He also provided the following rationale, based on the proof by contradiction (I do not have the original wording at hand, but I believe I can paraphrase the idea, which I am interested in.)

Let's imagine one heavy body and one light body. Suppose that heavy bodies fall faster than light bodies, as almost everybody believes. Connect the two bodies so that they form another body. This resulting body is heavier than the original heavy body, so according to the assumption it should fall faster. But on the other hand, the original heavy body is being inhibited in its fall by the connected light body, since this "wants" to fall with less rapidity. Due to this inhibition, the heavy body part should fall with less rapidity than it falls normally alone. So we arrive at contradiction, and the only way to resolve it is to reject the assumption. Instead, all bodies fall with the same rapidity (acceleration).

What do you think of this argument - is it valid, or no? For what reasons? It is very compelling at first, but on the other hand, should not the law of free fall be experimental law, rather than logical necessity? If so, where is the problem with the reasoning?

• Do you have a citation for the original: when did Galileo propose this argument? – WetSavannaAnimal Nov 24 '14 at 2:08
• @WetSavannaAnimalakaRodVance I believe it is in Two New Sciences (Wikipedia link which points to various texts of it). – tfb Feb 4 '18 at 12:14
• The implicit assumption of Galileo is that inertial mass equals gravitational mass, which needs to be justified experimentally. In other words, it is assumed that the "mass to mass ratio" is a universal constant for all objects. Then we choose a unit system to make that ratio equal to $1$. – Zhuoran He Feb 4 '18 at 15:57
• – Shing Feb 9 '18 at 12:12

What happens if we instead consider a pair of charged bodies with different charges and replace the constant gravitational field with a constant electric field in the vertical direction? Suppose that both charged bodies are attracted by the ground and that no gravitational field is present.

In Galileo's reasoning no description of the gravitational interaction is actually provided, so gravitational field could be replaced by the electric one.

Along Galileo's reasoning should we conclude that the charged bodies will reach the ground simultaneously? It seems so.

It would be generally false, obviously, also because what happens also depends on the inertial masses of the bodies, that play a role but are not mentioned. So, in my opinion Galileo's reasoning is untenable.

• Good point, the argument is not valid for acceleration due to electrostatic field. But I do not think this invalidates Galileo's reasoning. It assumes that acceleration in free fall is a function of the mass only, which if we translate to electrostatic accelerations as "acceleration is a function of the charge only" is not satisfied. This may be the reason the rest of Galileo's argument does not lead to correct result. In this way, some character of gravitational interaction seems to be involved in the reasoning. – Ján Lalinský Jan 8 '14 at 16:26
• Notice that we have two notion of mass to consider: The falling bodies behave as we know because inertial and gravitational mass are identical, this is not mentioned in Galileo's reasoning. I do not think the reasoning is correct. You are interpreting a posteriori Galileo's "proof", in my view. You should not read what it is not written therein. However I know many many opinions about that "proof"! Some colleagues of mine think that it is obviously right! – Valter Moretti Jan 8 '14 at 16:38
• I do not understand your last comment. Galileo almost certainly did not use concepts of inertial and gravitational mass, but that does not make his reasoning obviously incorrect. In general theory of relativity there is only one kind of mass, and this is not taken against it, on the contrary, its incorporation into theory is considered as a great accomplishment by Einstein. Could you say where exactly goes above reasoning wrong? – Ján Lalinský Jan 8 '14 at 17:02
• You may be right that a interpret Galileo differently than he meant, but I am more interested in whether the basic idea is right than in what Galileo originally meant. – Ján Lalinský Jan 8 '14 at 17:03
• "Could you say where exactly goes above reasoning wrong?" Here: " But on the other hand, the original heavy body is being inhibited in its fall by the connected light body, since this "wants" to fall with less rapidity. Due to this inhibition, the heavy body part should fall with less rapidity than it falls normally alone." – Valter Moretti Jan 8 '14 at 17:16

Rapidity is usually taken to be synonymous with velocity. But using your definition, this is a very logical deduction. However it almost presupposes what is already known in order to arrive at it's conclusion. Because when you connect the two bodies, it is plausible that whatever mysterious quality that makes heavier bodies fall faster then flows into the lighter body, informing it of the total mass of the system. For things falling in air, this mysterious quality would be the ratio of the drag force to the gravitational force, and is transmitted through internal forces. For things in orbit, it is tidal forces. It is not self-evident that there should be no such mysterious quality in a vacuum, and is only determined by experiment.

• So if I understand you, your objection to the reasoning is that : it $presupposes$ that the light part inhibits the heavy part based on what would happen if the parts were falling independently? But this seems natural, especially if the heavy part is connected to the light part from below by a thin rope or metal rod. Why should thin rod change the tendency of the two bodies to move with different accelerations and thus pull each other ? – Ján Lalinský Jan 8 '14 at 14:58
• My objection is that the reasoning presupposes Newtonian (or Gallilean) physics. Because in a system where heavier objects do accelerate faster (in vacuum), Gallileo's reasoning says that the lighter object must somehow learn about the combined object's weight in order to accelerate more. If you say right off the bat that it doesn't, you are in effect assuming what you are trying to prove. – lionelbrits Jan 8 '14 at 20:42

Galileo's reasoning is basically :

Given the "heavier objects fall faster postulation" (presuming no other parameters), where $G(m_i)>G(m_j)$ whenever $m_i>m_j$:

$$\mathbb{I}:x_1(t)=G(t,m_1);x_2(t)=G(t,m_2)$$

Setting the ground as zero, as well as upward is positive.

The consequence of postulation $\mathbb{I}$ (Connect the two bodies):

$$x_{(1+2),\mathbb{I}}(t)=G(t,m_1+m_2)\tag{1}$$

Where the position of connected objects $x_{(1+2),\mathbb{I}}(t)<x_1(t)$. (heavier object falls faster).

Then Galileo postulates "connecting two - one light and one heavy - the connected object 'wants' to fall with less rapidity" ($x_1(t)$ represent the position of lighter disconnected object) :

$$\mathbb{II}:x_{(1+2),\mathbb{II}}(t)=\lambda[x_1(t)-x_2(t)]+x_2(t)$$

Where $\lambda$ is some scaling parameter, satisifying : $0<\lambda <1$, such that the connected object's $x_{{1+2},\mathbb{II}}(t)$ always somewhere between $x_1(t)$ & $x_2(t)$. Since, $x_2$(the disconnected heavier object) is the upper bound and the lower bound for $x_{(1+2)\mathbb{I}}$ and $x_{(1+2)\mathbb{II}}$respectively; Galileo claims therefore there is a contradiction.

Galileo's argument is kind of "valid" (but kind of a cheat to me), since the postulation $\mathbb{II}$ is actually a premise - You can't derive postulation $\mathbb{II}$ from postulation $\mathbb{I}^\dagger$, as well as there is no particular strong reason for either postulations at that time(both gives us contradicting consequences).

Now what's left is how much we can justify Galileo's postulation $\mathbb{II}$:

If we further assume: A.) $G$ has no other parameters (such as charges) B.) $G$ is linear over mass $m$: $G(t,m)=m*G(t)$: then simply by adding $x_1$ and $x_2$ from the "heavier object fall faster postulation $\mathbb{I}$", we have : $$x_1+x_2=(m_1+m_2)G(t)\tag{2}$$

While based on $(1):$

$$x_{(1+2),\mathbb{I}}(t)=G(t,m_1+m_2)=(m_1+m_2)G(t)<x_1+x_2\tag{3}$$

plug $(2)$ into $(3)$, then we have:

$$(m_1+m_2)G(t)<(m_1+m_2)G(t), \forall t>t_{releasing}$$

Therefore, contradiction. Since postulation $\mathbb{II}$ plays no roles here, then postulation $\mathbb{I}$ is wrong (or assumption $A$, $B$).

Therefore, Galileo's reasoning is valid under a few presumptions. Where $G$ (and $\lambda$) exactly has to be determined by experiment.

$\dagger :$ At least I can't think of one. If you happen to know one, please do tell me.