# Falling, Falling, Falling…

## Historical walk-through…

It’s been a long time since Newton developed the inverse-square law that governed how massive bodies affect one another in space. Kepler and Galileo’s observations supported Newton. Probably the most important single work ever published in the history of Physics, Philosophiæ Naturalis Principia Mathematica explained the wonders of what Physics can do, which lead to further significant developments in the fiel. His theory explained most of the observations, but failed to account for a few (like Mercury’s precession of orbit, where Newton’s theory tells that the orbit stays at the same position). Then came General theory of Relativity, where Einstein unified light and gravitation. You ask me now, “You said Occam’s Razor can be used. Newton explained a way more simpler than Einstein” Of course. Remember how physical laws work? If we get a theory that accounts for unexplained observations, then it’s better to follow it, regardless of the complications involved. Einstein’s theory helps cosmology, astrophysics, particle physics, etc. That doesn’t mean Newton is wrong. We can still use his theory to launch a satellite into orbit, predict tidal effects caused by the moon, etc. Anyways, Newton’s theory is more than enough for our explanation today…

## All objects fall at the same rate..!!!

You may have watched Derek’s (Veritasium) video on “Falling”. He gives two balls (of different masses) to the people roaming around, tells them to drop the balls simultaneously. People realize that their prediction that heavier objects fall faster is wrong. Then, he gives the common explanation that inertia manages to cancel out the force of gravity on each ball.

Well, this can also be confirmed by an analogous experiment done by Galileo (the coin & feather) Now, I can support him with Newton’s law itself. If you remember correctly, Newton also developed his laws of motion (especially the one that superseded Aristotle, the second law) that a force can cause a change in velocity of the object, given by $F=ma$ (ignoring rockets, mass-energy, etc.). Gravity, being a force does produce acceleration. Earth & Planets, Sun & Stars, every celestial body accelerates, as they’re being influenced by gravity. Let $M$ & $m$ be masses of both objects separated by a distance $r$. Equate both forces and you’ll find the acceleration.

$\frac{GMm}{r^2}=ma\implies a=\frac{GM}{r^2}$

This is what we call the acceleration due to gravity, $g=9.8\ \mathrm{ms^{-2}}$. So, this conveys that all objects (keeping aside, the air resistance) fall at the same rate.

## Umm… Sorry. But, it’s not..!!!

There are a couple of things to be noted in the pedagogy above.

• First of all, $g$ is definitely a variable (as a function of height), and the $9.8$ is just an approximation. It’s not a constant after all.
• We have equated gravitational mass along with inertial mass (so called Equivalence principle) and finally cancelled both.
• A small note – What we mean by “same rate” is that reaching the ground at the same “time”

But, these don’t have anything much to do with the fact that all objects don’t fall the same rate”.  Because, the fact hides behind our Earth’s mass. It should be contemplated that the gravitational force depends on mass of both the objects. In this case, when we drop a ball, we neglect that the Earth too accelerates towards the ball. It’s very negligible though. Else, Newton’s third law would go wrong. So, it has to accounted. The Earth exerts a force on the ball which accelerates it, and as a consequence, the ball exerts a force back on Earth. Let’s say that the mass of ball is $10\ \mathrm{kg}$. The acceleration of Earth towards it, would be

$a=\frac{F}m=\frac{98}{5.98\times 10^{24}}$ $=16.38\times 10^{-24}\ \mathrm{ms^{-2}}$

On the order of negative 24’s… Eh? Of course. But, here’s where the situation totally changes. The acceleration of Earth towards the object depends on the object’s mass. Since the Earth is very large, to magnitudes of the order of 20s, we just don’t think of Earth’s frame of reference. We just approximate Earth to be stationary. When falling, the objects accelerate towards their common barycenter (center of mass, to be specific). Now, try comparing this experiment with different masses – a lead shot, a ball, a bus or let’s say, an Empire State building (It’s just imagination, we’re free to do anything). For the last two, we’d obtain a different result for acceleration, which means that Earth would accelerate towards the latter soon, when compared to the former. “Sooner” means “less time for the object to reach ground”. So, “It’s not really correct”. We have to synthesize from the center of mass frame…

In reality, the acceleration of the objects towards the center of mass, depends on total mass of objects (Reduced mass, or effective mass to be specific).

## Is it wrong to approximate then?

No, it’s not wrong. It’s our best weapon for most of the situations where nature plays all her clever tricks around. Just that the phrase breaks down when it comes to larger objects. Take Mars for example. What would happen if Mars were to fall towards Earth? I can guarantee you that Mars would take less time to reach Earth, when compared to balls.

So, we have to phrase it this way…

Objects fall at the same rate (regardless of air friction), as long as the mass of the falling object is too small when compared to the other. Or, you can say, Two massive objects are attracted to each another towards their common barycenter.

### Try this:

Now, here’s an interactive simulation called My Solar System. You can play a lot with it, as it plots every single motion once the time starts.

Once you’re done playing, target the sun with a planet. Try changing the mass of the planet and see how the sun responds. You can notice that the distance covered by Sun is different for different masses of the planet.

• When a planet falls “into” a star (about 20 times more massive)

• The same planet, just 5 times more massive than it was before…

The bit-rate is quite screwed up in this GIF image. In reality, the time taken by the planet to fall into the star in second case is somewhat less than it took at the first (Well, you can still note the distance the star moves in both cases)

Originally inspired by this post written by David Zaslavsky (a grad student) at Physics Stack Exchange.