# Gravity and free fall

In Wikipedia it's stated that "[..] gravity, is the natural phenomenon by which physical bodies appear to attract each other with a force proportional to their masses".

Then I found many examples regarding free fall and gravity, such as "a hammer vs feather", "an elephant vs a mouse", etc. They all say that in the absence of another external force (like air-friction/resistance), the hammer, the elephant, the mouse and even the tiny feather, will all fall freely with the same speed and at a constant acceleration (like $1g$).

Ok, I think I can understand that, it makes sense.

Now my question regarding these "experiments":

• let's suppose that we have an object $E=$ Earth (with its mass ~$6 \times 10^{24}$ $kg$)
• let's suppose that I have other 3 distinct objects: $O_1=$ a planet with a mass $5.99 \times 10^{24}$kg, $O_2=$ an elephant and $O_3=$ feather.

If we imagine that the all 3 objects are situated above the Earth at a distance $d= 250$ miles, and if we suppose that there is no air-resistance or any other external force to stop their free-fall, then what will happen with ours objects? Which will fall first, second, third?

My understanding is that the $O_1$'s inner force will fight to attract the Earth toward it. The Earth will do the same thing but, having a mass just "a bit" larger than $O_1$, the Earth will eventually win and, will eventually attract the $O_1$ toward it ("down" to Earth). The same thing will happen with the elephant and with the mouse/feather, except that the elephant having a larger mass than the mouse (or feather), will fight just a bit more than the others and will decelerate its fall with $0.00000...1\%$ (let's say) comparing with the mouse/feather (which seems almost the same thing if we ignore few hundred decimals).

• Possible duplicate: physics.stackexchange.com/q/3534/2451 – Qmechanic Feb 5 '13 at 0:26
• In this question, he is assuming that all objects are dropped together, and asking what will happen, and what will fall first. See my answer, they will all hit the Earth at the same time. If however a separate experiment is carried out for each mass, as done in the question you have linked, you will obtain a different time for each mass. – Kenshin Feb 5 '13 at 0:29
• I think you are wrong to think in terms of some objects or forces fighting and winning or losing. If O1 has identical mass to earth, there's no sense in which a stalemate ensues with neither moving. For each case (including the feather) The question of which object moves is only a matter of where you choose to have your reference point. – RedGrittyBrick Feb 5 '13 at 19:02

Assuming that the three objects you speak of are point particles initially positioned pretty much right next to each other, then all three objects will hit the earth at the same time.

From newton's law of gravitation, we have:

$F = \frac{Gm_1m_2}{r^2}$ and where $m_1$ is the mass of the Earth, and $m_2$ is the mass of an object away from the Earth (e.g. feather, or O1).

To obtain acceleration of the object (e.g. feather), we divide by mass, so:

$acceleration = \frac{Gm_1}{r^2}$

As we can see, the acceleration of the feather only depends upon the mass of the Earth. Therefore, the feather and object O1 and the elephant will all accelerate towards Earth at the same rate.

However,

O1 has a very large mass, and this will cause the Earth to accelerate towards O1. But since all three objects were initially positioned in the same spot, and all three objects are accelerating towards Earth at the same rate, the three objects remain next to each other, and therefore when the Earth reaches O1, it will reach the feather and the elephant at the same time.

• Yes - if the objects fall together they will stay together. But if you did separate experiments the heavier object would touch down slightly earlier. You might need a really really big elephant to be able to measure the difference... – Floris Dec 7 '14 at 23:31
• You also forgot to mention that the three object will attract each other as well, such that by the time they hit the Earth they would all be sticking together, or rather to $O_1$. – fibonatic Dec 8 '14 at 8:39
• @fibonatic, yes my assumption was that the three particles start off together anyway. – Kenshin Dec 8 '14 at 17:07
• Interesting side note - experiments to confirm that there is no difference between "inertial mass" (as in $F=m\cdot a$) and "gravitational mass" (F=\frac{Gm_1m_2}{r^2}\$) are quite a field in their own right. AFAIK It has not been proven they are different- but that doesn't mean they are the same. – Floris Dec 10 '14 at 5:07
• @Floris, interesting point you raise. Many of us forget this fact and fall in to the trap of simply assuming they are definitely equal, thanks for your reminder. – Kenshin Dec 10 '14 at 5:54

I believe the Earths Inertia would prevent any of the other 3 objects from moving it. Their Gravity can not overcome the Earths inertia. Therefore The Earth stay's where it is and the 3 land at exactly the same moment.

• I think you are missing the point of the question - a "sufficiently heavy" object would move the earth up towards it. – Floris Dec 7 '14 at 23:33