# A Thought Experiment

## Why All Objects Fall Equally Fast

Use thought experiment to explain why all objects are falling equally fast.

"Derive" Newton's 2nd law in the process. $$F = m\cdot a$$

Let us discuss one of the most intriguing phenomena in physics that we can observe in everyday life: objects in free fall.

Most of us learn, or are told rather, that all objects are accelerated in the same manner ("fall equally fast") when released in Earth's gravitational field.

In other words, if we release any two objects at the same height and at the same time, they will remain next to each other while they are falling.

For many of us, this is counterintuitive:

We think heavier objects experience a larger force and, therefore, they should fall faster.

Let us look at the situation in more detail in a thought experiment (from German: Gedankenexperiment). Such experiments cannot necessarily be conducted in practise but they are nevertheless useful in that they tell us a lot about nature and how it may behave.

To simplify the matter, we will neglect air drag because one thing is for sure: a stone falls faster in air than a feather. To address the impact that shape may have on the motion of falling objects, we consider three identical objects, meaning identical in shape, size and weight, that are in free fall in a vacuum.

For now, we assume that these spherical balls of mass m are made of the same material, iron say.

Clearly, one would expect all 3 balls to fall equally fast when released at the same time with zero initial velocity. So far, so good!

Now, let us add another ball to the picture, having mass 2m. It could have twice the volume of the other balls, or the same volume, or any other volume. Let us say it has twice the volume.

Since the force onto ball 4 is twice as large as that on balls 1, 2 or 3, we could expect ball 4 to accelerate faster.

Is that so?

Assume all 4 objects are released at the same time and height, and with zero initial velocity.

Assume also that ball 4 is falling faster than balls 1, 2 and 3. Balls 1, 2 and 3 obviously fall equally fast since they are identical. Now imagine you were a massless person standing on ball 1 and observing what is happening around you.

You would always see balls 2 and 3 next to you while ball 4 would begin to move away from you in the downward direction, i.e. the direction in which you are moving.

Now imagine two massless people, one standing on ball 2 (Amy) and one standing on ball 3 (Bill).

While you are falling, Amy throws a rope over to Bill. She is holding on to one end of the rope.

Bill catches the other end of the rope.

Since there is a connection now, would you say that this makes ball 2 and ball 3 one object? If the answer is yes, this object has now mass 2m and it should begin to accelerate like ball 4 the moment that Bill catches the rope. This means that, all of a sudden, it would move away from you standing on ball 1, simply because Amy threw a massless rope over to Bill. This does not seem very plausible!

It seems more reasonable that Amy and Bill will not move away from you. They will not begin to move faster.

OK. The next thing that happens is that Amy is pulling on the rope so that Bill and Amy are approaching one another until ball 2 touches ball 3.

Would you say now that ball 2 and ball 3 form one single object?

If so, it should now begin to fall faster the moment the objects touch. Having mass 2m now, this single object should begin to accelerate like ball 4. This would look strange to you on ball 1. Until the moment balls 2 and 3 touch, Amy and Bill would be falling next to you, being at the same height at the same time.

The moment balls 2 and 3 touch, Amy and Bill begin to move faster than you! It is as if someone is telling all the little atoms in balls 2 and 3 to behave differently just because a couple of atoms at the surface of balls 2 and 3 touch one another.

That does not sound plausible either!

OK. Let's say that Amy also brought massless welding equipment with her. After balls 2 and 3 touch, she begins to weld both objects together.

Nobody would argue now that the two balls form one object. They are inseparable unless a substantial force is applied to separate them. Clearly, they now represent one object of mass 2m.

If Amy and Bill feel like it, they could even reshape the object they are on to have the same shape (and the same volume) as ball 4. Now, most definitely they would fall as fast as ball 4. But when exactly do we have one object?

When Amy has welded balls 2 and 3 together? If so, when is the welding process considered complete? Or do they need to bring their object into the same shape as ball 4? Can they be off by one atom being misplaced?

Remember this is all happening while you are watching the events from ball 1. You would observe that Amy and Bill begin to fall faster the moment Amy finishes the welding job, or the moment they reshape their object to be spherical and, hence, identical to ball 4.

In each case, just when the last atom is put in place to finish the welding or reshape the object, all atoms suddenly "decide" to move faster, and Amy and Bill move away from you.

I cannot help it: This still does not sound plausible!

Reflecting upon the above, we see that our discussion is not restricted to spherical objects made from iron. The same train of thought above applies to any material and shape. There is only one logical explanation that unravels the apparent paradox we encounter:

Ball 4 does not fall faster than ball 1 (or 2 or 3) to start with!

In other words:

All objects fall equally fast!

Yes, all objects fall equally fast!

They fall equally fast, independent of their chemical composition, shape, size or, most importantly, weight.

The acceleration is the same for all objects.

# Newton's Second Law

Since the gravitational force onto an object doubles as the mass doubles (twice the atoms, twice the interaction between object and Earth, hence twice the force), we see that the ratio F/m of the force and the mass remains constant, just like the acceleration did.

Therefore, we can write down a relation between acceleration of an object, its mass and the force acting upon it

(1) $$a=G\left(\frac{F}{m}\right)$$

where $G$ is an unknown function.

Equation (1) is what has come out of our thought experiment above. The simplest relation is obviously

(2) $$a=\frac{F}{m}$$

from which $F=m\cdot a$ follows.

(Footnote: Note that $a=c\cdot F/m$, with $c$ a constant, can be written as $a=F/m$ by rescaling mass. Someone has to define 1kg! Defining a meter, a second and a kilogram will define 1 Newton.)

Can we derive equation (2) of $a=F/m$, i.e. linearity, with another thought experiment?

Not really! As one learns in special relativity, equation (2) needs to be modified when an object moves very fast, at an appreciable fraction of the speed of light. Therefore, F=m*a is only an approximation for small velocities.

Nevertheless, this has been one powerful thought experiment!

# A Note on the Side

We could have made a similar argument that all objects fall equally fast by looking at an object of mass $2\cdot m$ that is slowly stretched until it breaks up into two pieces, each of mass $m$.

# A Note on the Side (1)

We could have made a similar argument that all objects fall equally fast by looking at an object of mass $2\cdot m$ that is slowly stretched until it breaks up into two pieces, each of mass $m$.

# A Note on the Side (2)

We could have made a similar argument that all objects fall equally fast by looking at an object of mass $2\cdot m$ that is slowly stretched until it breaks up into two pieces, each of mass $m$.

# A Note on the Side (3)

We could have made a similar argument that all objects fall equally fast by looking at an object of mass $2\cdot m$ that is slowly stretched until it breaks up into two pieces, each of mass $m$.

This time, the question is: why or when would the two objects of mass m begin to fall more slowly? When the final, arbitrarily thin connection between them is cut?

We come to the same conclusion as before:

All objects fall equally fast!

# Summary

In this lesson, we learned that:

1. Thought experiments can be very powerful.
2. Many phenomena in nature can be explained by "virtual" experiments that can often not even be realized in practise.

In partcular:

• All objects fall equally fast (when air drag is not present, i.e. in a vaccum).

• Newton's 2nd Law, $F=m\cdot a$, can be derived intuitively, meaning by pure thought, but it cannot be proven to be true. The latter has to do with the assumptions we need to make. (For example, we have assumed that the masses above can only change in a discontinuous fashion, i.e. when two of them form a new mass or when they are divided. Also, we discussed our example within the framework of Newtonian mechanics only.)

• Sometimes, very little mathematics is required to understand nature or to make predictions about nature.