# Objects of Different Density in Freefall

## Main Question or Discussion Point

Hey all,

I was hoping to settle a debate once and for all.

Suppose you've got two identical objects, in our hypothetical situation they were mini-vans. One of these is empty and the other is filled with something heavy like lead so that it is 100x the weight of the empty one. Both objects therefore have the same aerodynamic properties, but one is much heavier and therefore much more dense than the other.

If we were to release these objects from the same height at the same time (in an environment such as the earth's atmosphere, not a vacuum) would they hit the ground at the same time?

Since the rule of thumb that "all objects fall at the same rate" has the qualifier that the objects must be in a vacuum, I am tempted to believe that the denser object would fall faster. At the very least, I would think that it has a higher terminal velocity.

To attempt to prove this, I proposed the following thought experiment:

Suppose you've got two sheets of paper that you fold into identical cubes. One of the cubes is left empty while the other is filled with something rather heavy. We notice significant air resistance in paper, so I thought this would be a suitable example. The counterargument proposed was that the paper would deform from the air hitting it and therefore change shape. I don't believe that such a change would be significant, but I think that even if I used something slightly more sturdy (like balsa wood) the results would speak for themselves.

Thanks!

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Simon Bridge
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Imagine two ping-pong balls - one you fill with, say, a jelly (via a syringe perhaps) and the other is left empty. You seal both of them so they are air-tight. Will the fall at a different rate?

Imagine two large foil bags - one you fill with compressed air and the other with Helium gas (both at room temperature) ... do they still fall at the same rate?

We would expect the air, like any fluid, to exert a buoyancy force proportional to the volume of air displaced. This is, after all, how hot-air balloons fly.

For the situation where the objects are not air-tight it can get complicated. Picture repeating the experiment in water. Still, in general, the heavy object sinks faster.

In a way "falling" is the situation where the body is "sinking" with insignificant buoyancy.

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Terminal velocity occurs when air resistance equals the weight of the falling object. So I would say the heavier object will have a higher terminal velocity than the lighter object because air resistance depends on speed.

Simon Bridge
Homework Helper
@Emilyjoint: that too :)

If you have mass falling in a vacuum, say on the moon, If one massed 1 million kg and the other 1 gram, The regular word is they would hit the surface at the same time. But wouldn't there be a slight difference due to the gravity of the large mass pulling the moon to move ever so slightly and therefore landing maybe attoseconds sooner?

For instance, if a body the mass of the moon was near the gravity well of the moon, wouldn't it hit faster than a one gram mass?

Simon Bridge
Homework Helper
@Litup - different, if related, question.
Note: technically - in Newtonian gravity - all three masses are gravitating towards their common center of mass - they will all hit that at the same time in the reference frame of the center-of-mass. (Someone may want to do the math on this - though 3-body Newtonian mechanics is tough! You want the surfaces of the small bodies to be the same distance from the surface of the Moon or the bigger radius will hit first.)

Shall we bring in that there are many influences besides the two masses in the experiment on the Moon (and the masses) which could throw off the experiment on an attosecond scale? How about general relativity? Everything we are talking about here is an approximation.... a model. We are discussing predictions in different models. What happens for real is what you measure happening.

@Litup - different, if related, question.
Note: technically - in Newtonian gravity - all three masses are gravitating towards their common center of mass - they will all hit that at the same time in the reference frame of the center-of-mass. (Someone may want to do the math on this - though 3-body Newtonian mechanics is tough! You want the surfaces of the small bodies to be the same distance from the surface of the Moon or the bigger radius will hit first.)

Shall we bring in that there are many influences besides the two masses in the experiment on the Moon (and the masses) which could throw off the experiment on an attosecond scale? How about general relativity? Everything we are talking about here is an approximation.... a model. We are discussing predictions in different models. What happens for real is what you measure happening.
What if the masses are on opposite sides of the moon? Major mass on the left, minor mass on the right?

Simon Bridge
Homework Helper
They still all gravitate towards their common center of mass.

To be authentic, the experiment should be viewed from the Moon reference frame.
You should have a better time doing the math but recall that you cannot use the short-distance approx for the two small masses.

Perhaps, try putting the Moon at the origin, mass M, radius R, and let the masses be small enough to approx as points with m1 at -(R+h) on the x axis, and m2 at +(R+h) i.e. so they are each the same initial height from the Moon's surface... work out the acceleration as a function of time for each m1 m2 and M and look for the time when the distance of each m1,2 to the Moon is R. Compare.

Careful of approximations - m1 and m2 are 2(R+h) apart.