# I How strong is gravity?

#### zuz

Hypothetically, if the universe was completely empty, and two apples were 10,000,000,000 light years apart, would their gravity be strong enough to eventually bring them together?

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#### Nugatory

Mentor
What do you think and why? Can you calculate the force between the two apples if you make reasonable assumptions about their mass?

#### zuz

Me! No. I can't. That's why I asked PF.

#### Ibix

It's difficult to know how physics would behave in a completely empty universe, since it isn't clear that our physical law would apply to such a universe. That's not what it produced, after all.

However, making the assumption that we could use our physical law, the answer is that it depends on things you haven't stated. The gravitational force between two bodies goes as $1/r^2$, so never goes to zero as far as we are aware. However, an object which has escape velocity is one going fast enough that the slowing effect of gravity isn't enough to ever stop it. The escape velocity against the gravitational field of an apple is incredibly low. If the apples aren't stationary or moving together, then they'll probably never come together.

#### Nugatory

Mentor
Me! No. I can't. That's why I asked PF.
Google for "Newton Law of Gravity", then try calculating it for yourself.

#### HallsofIvy

Homework Helper
"Newton's law of gravity", referred to by Nugatory, says that $$F= \frac{GMm}{r^2}$$ where F is the force between two objects of masses M and m with distance r between them. G is a "universal" constant, its specific value depending on the units used. In the "MKS", "Meter-Kilogram-second", system, it is $6.67\times 10^{-11}$ Newton meters-squared per kg squared.

So, in this case, where r is 10,000,000 light years, r squared is going to be very, very, large, the gravitational force very, very, small but still non-zero. With no other masses in the universe, the "apples" will each apply a very, very, small force on the other so will start moving very, very, slowly toward each other. That force will slowly increase so the speeds will slowly increase until, after millions of years the two apples will smash together creating the only apple sauce in the universe!
(And leaving us to ask "where are the apple trees?)

Cool. Thanks.

#### DaveC426913

Gold Member
Correction:
So, in this case, where r is 10,000,000 light years, r squared is going to be very, very, large, the gravitational force very, very, small but still non-zero.
...10,000,000,000 light years...
...very, very, very large...
...very, very, very small...

#### FactChecker

Gold Member
2018 Award
Remember that there is an escape velocity that allows one mass to go infinitely far away from another mass. In the case you present, the escape velocity would be extremely small.

PS. I can't resist throwing out an ignorant, amateur thought here. The forces and escape velocity may be so tiny that the uncertainty principle becomes relevant.

EDIT: I didn't notice that @Ibix had already mentioned the escape velocity in an early post (#4).

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#### phinds

Gold Member
PS. I can't resist throwing out an ignorant, amateur thought here. The forces and escape velocity may be so tiny that the uncertainty principle becomes relevant.
Yeah, but since the op posits a magical universe (apples without apple trees as just one example) we can just decide that the HUP doesn't exist there.

#### Janus

Staff Emeritus
Gold Member
That force will slowly increase so the speeds will slowly increase until, after millions of years the two apples will smash together creating the only apple sauce in the universe!
(And leaving us to ask "where are the apple trees?)
As mentioned by Ibex, the escape velocity for an apple is really, really, really small. This, in turn, means that the collision speed between the two apples, even if they start 10 million light years apart, is going to be of similar magnitude. Not even enough to produce any significant bruising, let alone applesauce.

#### willem2

PS. I can't resist throwing out an ignorant, amateur thought here. The forces and escape velocity may be so tiny that the uncertainty principle becomes relevant.
The escape velocity at 10^10 light year is 4.2*10-16m/s (assuming a 0.1 kg apple)
You're still fine with the uncertainty principle for the radial speed, but what about the tangential speed? If that is even a very tiny fraction of the escape velocity of 4.2*10-16m/s the apples will never come near each other, but go in orbit.

#### Ibix

The escape velocity at 10^10 light year is 4.2*10-16m/s
I get a somewhat smaller figure - about 3.8×10-19m/s.

#### willem2

I get a somewhat smaller figure - about 3.8×10-19m/s.
Yes, you're right.

#### DaveC426913

Gold Member
As mentioned by Ibex, the escape velocity for an apple is really, really, really small. This, in turn, means that the collision speed between the two apples, even if they start 10 million light years apart, is going to be of similar magnitude. Not even enough to produce any significant bruising, let alone applesauce.
Does this follow?

I guess I'd have to calculate their final velocity after falling 10 million (billion!!) light years.

#### Janus

Staff Emeritus
Gold Member
Does this follow?

I guess I'd have to calculate their final velocity after falling 10 million (billion!!) light years.
Escape velocity is the speed that an object would have to have at some distance "R" from the center of mass, in order to never fall back. So imagine you have your two in-falling apples, and instead of hitting, they started with just enough sideways motion to just skim past each other with their centers 8cm apart. At that point, they would have almost exactly at the same speed as they would have had in the moment just before they touch, if they had fallen directly towards each other

If you assume that their velocity at that their velocity at that moment is much larger than escape velocity at that distance, then they would after separation continue to separate to an infinite distance, rather than just back out to their starting distance. Escape velocity is not only the minimum speed needed to escape to infinity, it is also the maximum speed for an object dropped (starting at rest) from an infinite distance.

#### Ibix

Does this follow?
If their velocity is initially zero at near enough infinity to make no difference, their velocity at impact will be the escape velocity from their center-to-center distance at impact.

Taking 0.1kg, 0.1m apples for convenience, I get about 10 microns per second.

#### DaveC426913

Gold Member
TIL.

So, an object, initially at rest WRT Earth, falling from infinity, will be going 11.2 km/s at impact (sans atmo and all other confounding factors)?

#### Ibix

Yes. As A.T. says, you can see it from conservation of energy, insisting that KE=0 at infinity tells you that at ground level it must be equal to the change in gravitational potential energy. And that's true going up or coming down - I've made no statement about which state was the first one.

#### DaveC426913

Gold Member
Yeah.
TIL.

So, an object, initially at rest WRT Earth, falling from infinity, will be going 11.2 km/s at impact (sans atmo and all other confounding factors)?
I just realized that this is trivially true, since it's reversible.
If you send an object up from Earth at anything less than escape velocity, it will fall back (eventually), even if it takes 10,000,000,000 years. And its return velocity will be the same as its initial velocity.

Falling from infinity is merely the second half of that trajectory.

#### kuruman

Homework Helper
Gold Member
Just out of curiosity, I calculated the time it would take for the two apples to collide if they were released 10 million light years apart. From energy conservation, one gets$$\frac{dr}{dt}=-\sqrt{\frac{Gm}{r_0}\left(\frac{r_0-r}{r}\right)}$$which upon integration gives the time$$T=\frac{\pi}{2}\sqrt{\frac{1}{Gm}}r_0^{3/2}=6\times10^{32}~\mathrm{years}.$$By comparison, the age of the universe is thought to be $14\times 10^9~\mathrm{years}.$

On edit: Although the calculation for $T$ is correct for an initial separation of 10 million light years, it is not directly relevant to OP's posted distance of 10 billion years. To find the time for OP's distance, mutiply $T$ shown above by $1000^{3/2}.$

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#### zuz

Wow. You guys are way, way, way over my head, but it makes for some interesting reading(what I can understand of it anyway) Thanks.

#### DaveC426913

Gold Member
Just out of curiosity, I calculated the time it would take for the two apples to collide if they were released 10 million light years apart.
<rant>
OHMYGOD.

Billion!
The OP's example was 10 billion light years separation!

(see post 8 if you think I'm overreacting...)
</rant>

#### kuruman

Homework Helper
Gold Member
<rant>
OHMYGOD.

Billion!
The OP's example was 10 billion light years separation!

(see post 8 if you think I'm overreacting...)
</rant>
[mea culpa]
Silly me! I looked at the wrong post when I counted zeroes. Goes to show why powers of 10 are better for expressing large numbers even in cases where the number of zeroes is no larger than the number of fingers on one's two hands. I edited my previous post to address the issue.
[/mea culpa]

"How strong is gravity?"

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