# 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? Nov 24, 2014 at 2:08
• @WetSavannaAnimalakaRodVance I believe it is in Two New Sciences (Wikipedia link which points to various texts of it).
– user107153
Feb 4, 2018 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 $1$. Feb 4, 2018 at 15:57
• Feb 9, 2018 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. Jan 8, 2014 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! Jan 8, 2014 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? Jan 8, 2014 at 17:02
• Well this point in particular: "the original heavy body is being inhibited in its fall by the connected light body, since this "wants" to fall with less rapidity". It is a "non sequitur". What does "wants" mean there? It is not possible to achieve any conclusion like Galileo's one, in my opinion, without a precise notion of dynamics and interaction, for instance the Newtonian one (but not necessarily that). Jan 8, 2014 at 17:57
• The rules of the game are not defined and the reasoning relies upon a rhetorical argumentation only. Jan 8, 2014 at 18:00

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 ? Jan 8, 2014 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. Jan 8, 2014 at 20:42
• The argument seems to be that Galileo wanted to show that a contradiction follows if we assume the hypothesis is true. But to derive the contradiction he declares it false when he says the heavy body is being inhibited. If he wanted to assume the hypothesis was true he should have assumed that, indeed, the pair magically falls faster once they become connected.. Oct 12, 2019 at 14:10

The argument seems valid and in fact rather evident when considered as a comment on rigidity: if two lumps of clay accelerates differently under gravity, how do you explain that aggregating them together gives an object that keeps its rigidity when falling (instead of being subject to the equivalent of destructive tidal forces)? Since this observation is true for all physical bodies, not just clay of course, we would need a universal theory of cohesion that seems very much non-trivial.

You can see in the timeline of solid mechanics that Galileo Galilei is in fact an important contributor to the theory of cohesion, and so it seems natural to me that this sort of consideration let him recognize that since falling objects do not experience any specific structural stress, it must mean that the free fall acceleration of their parts must be the same, whatever the parts are, and by natural extension that the free fall acceleration of all objects must be the same.

• You are assuming that the falling object does not experience stress forces pulling it apart. The argument I asked about does not assume that. Jul 12 at 15:16
• You are right - I just think that the (historical) argument cannot be that deconnected from observation, and does have some kind of experimental (or intuitive) basis. The subject could certainly be explored further... Jul 12 at 18:04

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.