# Do heavy and light objects fall at the same speed?

### High School Physics Class

I learned in school, just like everyone else, that more massive objects do not fall faster. In the last few years I did some thought experiments that have left me wondering. Recently a friend of mine decided to put my ideas through all the gravitational formulas, and it seems like my High School physics teacher was flat wrong!

### The Popular Experiment Is Useless

Drop two objects, one heavy and one light, and observe that they do indeed fall at the same rate. This experiment is fatally flawed, even ignoring wind resistance.

 Dropping dissimilar objects: which distance to use?
First of all, consider that the center of gravity (COG) of a larger object is farther from its surface than the COG of a smaller object. In the experiment, if the objects are held the exact same distance from the Earth, but the measurement is made to the outside, to the bottom of the objects, then the larger object's COG will be farther from the earth and thus experience less gravity. (This is a gravitational law: the acceleration due to gravity drops off by the square of the distances between the objects.) If the objects are held so that their COGs are the same distance from the Earth, they have different distances to fall.

 Each bit of subatomic mass in two objects attracts every other bit.
However, looking at the COG is still not complex enough to understand gravity fully. While it is convenient to consider the COG of an object as the point upon which gravity exerts itself, gravity really is an attraction between each and every bit of subatomic mass in the universe to each and every other bit of subatomic mass in the universe. To accurately calculate the forces due to gravity between two objects, one would have to add the vectors of attraction between all the mass particles in them. How do you calculate the force due to gravity when you are inside the Earth, for example, or allow for an object changing shape as it falls because of gravity's increase as it moves closer, when simply using the COG as your reference?

The next most obvious reason that the experiment is flawed is that a human could never hold the objects at the same height, drop them at the same moment, or detect that they struck the Earth at the same moment. It's possible that even the most advanced equipment could not measure a difference in when the objects land. This does not mean that the difference is not there, simply that we must turn to experiments and formulas to provide the answer.

Last, the most important yet least obvious reason that the experiment is flawed isn't a problem with the experiment. It's that the experimenters only look at half of the equation! Let's handle the other problems before dealing with this one.

 Dropping... Fruit? O has more force exerted on it, but this force is required to move more mass. G has less force exerted on it, but this force is moving less mass. O and G thus receive the exact same acceleration.
 Dropping...The Moon? M has more force exerted on it, but this force is required to move more mass. M receives the exact same acceleration that O and G did.

### A Better Experiment

Let's do some thought experiments. To avoid the problems mentioned, we are going to imagine objects that are uniform in every possible way, and "test" them in a vacuum. O is an orange-shaped, orange-sized, orange-massed object. G is an orange-shaped, orange-sized, grape-massed object. O is placed in a vacuum. It is held an exactly known distance, d, from the floor and is dropped by special equipment that times its descent. The same exercise is repeated for G. The expected result is that both objects will traverse d in the same amount of time, t. The Earth does indeed impart identical acceleration to each object, and this is a natural conclusion to make.

Now, repeat the experiment once again with a new object, M. M is an orange-shaped, orange-sized, Moon-massed object. The expected results are that it will also traverse d in t, just like O and G. There's a lot more force acting on it, but there's a lot more mass being acted on.

### A New Frame of Reference - The Moon

The experiments with objects O and G are this time repeated on the moon. The acceleration, a, will now be approximately 1/6th what it was on the Earth and t will be 6 times as long—the moon has approximately 1/6th the gravity.

Again a new object is introduced: E is an orange-shaped, orange-sized, Earth-massed object. The expected result is that E will cover the distance in the same amount of time as objects O and G. Do you see the problem? The very large difference in mass allows us to see the part that we missed before: E also accelerates the moon toward it as well! All this time we've only been considering the acceleration of the object being "dropped."

If you like, to eliminate tidal effects and differing radii of our frames of reference, the above experiments could be repeated all over again, only with some changes. In the first series of experiments, instead of dropping objects toward the Earth, drop them toward E. In the second series of experiments, instead of dropping objects toward the Moon, drop them toward M. The final result (if heavier objects really do fall at the same rate) is that M "drops toward" E at 6 times the rate that E "drops toward" M. They are the same two objects, so this is clearly incorrect!

### Thinking In A Box?

Now to return to the most serious problem I mentioned earlier, about only looking at half the equation.

It's easy to think about objects as falling "toward" the Earth—the fixed, immovable, force-generating Earth. Objects are seen simply as passive receivers of this great force. Even the language of the experiment underscores this conception:

• Raising an object instead of moving apart two objects
• Height instead of distance between
• Holding an object up instead of holding two objects apart
• Dropping an object instead of allowing the objects to move freely
• Falling instead of accelerating or moving toward
• Rate of descent instead of closing speed

In the experiments, the Earth and the object being dropped are in reality being held apart from each other. When the force holding them apart is removed, both objects then accelerate toward each other. Each object's acceleration is proportional to the mass of the other object. The sidebar shows this with red force arrows: the acceleration imparted to the orange object is related to the teal object's mass, and vice versa.

 Acceleration is imparted to each object, including E

### Forgotten Forces

Here's the problem. The Earth isn't fixed, any more than an orange we drop is fixed.

• The Earth has more mass, so it accelerates an object toward itself faster than the moon accelerates an object toward itself.
• An orange has more mass, so it accelerates the Earth toward itself faster than a grape accelerates the Earth toward itself.
• Both the Earth and the orange are being acted upon by the other.

The force exerted by the orange is of course minuscule. It may be so small that any experiment we can actually devise and carry out would not show the difference between it and a grape falling because we do not have equipment advanced enough to detect it.

### Making It Even More Complicated

I'll also note that everything is complicated by taking into effect tidal forces. Tide is just a simple way of saying that objects like oranges and the Earth are not perfectly rigid and that the force due to gravity falls off by the square of the distance. The Earth is constantly being "squished" around by the moon. This is a source of friction and is actually causing the Earth to slow its rotation and the moon to move away from the Earth. If the process is allowed to continue, the Earth will eventually become tidally locked with the moon, and the moon will at that point begin moving closer to the Earth, and eventually crash into it! The moon is already tidally locked with the Earth—from the Earth you always see the same face of the moon no matter where you go or when it is.

What does this have to do with dropping objects? Well, the net effect of gravity on an object is one of stretching. An object changing shape while it falls can change the distance between its COG and the side closest to the frame of reference, changing the results of the experiment. In order for all these experiments to perfectly get the expected results, all the objects would have to be perfectly rigid.

 Forgetting Something? Whatever happened to the person's accleration?

### The Clincher

I finally came up with one last thought experiment that put it all together for me.

Repeat the original, classic experiment, with some new objects. Person is an Earth-shaped, Earth-sized, human-massed object. It is the new frame of reference. Drop two objects, one heavy and one light, toward it, and see if they fall at the same rate. For the heavy object, use E. The light object will be M.

Do you see that E will indeed "fall toward" Person at 6 times the rate that M will "fall toward" Person? Both E and M do indeed receive the exact same acceleration, just like in every other experiment we did. But something's missing, isn't it? Do you also see why the experiments must be conducted separately?

### The Conclusion

Heavier objects do fall faster.

### Spelling It Out

If you're still convinced that light and heavy objects fall at the same rate, then consider what you mean by fall.

All objects really do accelerate toward the Earth at exactly the same rate, but acceleration of the object is not the only thing that's happening. There is also the acceleration in the opposite direction of the object toward which it is being dropped.

If one simplistically defines "falling rate" as the absolute speed of the object being dropped (when it is allowed to move freely by the force of gravity), then objects do fall at the same rate, no matter their mass. I am not denying that!

But if one more accurately defines "falling rate" as the closing speed between two objects (when they are allowed to move freely by the force of gravity), then objects do not fall at the same rate. The closing speed must involve the acceleration and movement of the other object too. It doesn't matter if it's big. It doesn't matter if you're standing on it. It doesn't matter if the forces are so minuscule and the effect so minor that current technology can't measure it. If the effect is there at all, then objects do not fall at the same rate. They fall at immeasurably close, but unequal, rates.

It's really impossible to determine the absolute speed of any object. There's no spacegrid—like the holodeck's hologrid in Star Trek—to look at. Is the object moving in relation to fixed space and if so, how fast? Current science cannot answer this question. Anywhere we could possibly go to do experiments like these will have other considerations. Some of them are: the rotation of the Earth (causing Coriolis force because of its shape); the revolution of the Earth about the Sun; the Sun's rotation; the mass, positions, velocity, and spin of the other planets; our solar system's orbit in this galaxy; our galaxy's movement in the universe.

Please do hesitate to write me unless one of the following is true:

• You are a physics professor or a scientist.
• You have thought about it for at least a few days.
• You've got some relevant facts or web references.
• You have a spelling, grammar, or style correction for me (and you know your stuff).

I plan on eventually showing all the formulas for those who'd like to see them, but I thought this was enough for one day's work.

Happy thinking!

This is an original work by Erik Eckhardt.
All contents of this page are © Erik Eckhardt 2001. You may not copy or reproduce this information, in whole or in part, anywhere or in any form without explicit written consent.