Does a Bowling Ball Fall to Earth Faster Than a Planet of Similar Mass?

[Robert Harris]

I'm looking for knowledgeable help to resolve an argument between myself and someone else.

A bowling ball is positioned, say one hundred thousand miles from the Earth, motionless, relative to the Earth. It then begins to fall, finally crashing into the planet after some period of time.

Another planet, of similar mass and dimensions of the Earth, is positioned at exactly the same distance as the ball, also motionless. The two planets begin to pull together and after some period of time, collide, spoiling a lot of people's day.

Would it take the bowling ball longer to collide with the Earth than the other planet, the same amount of time, or less time?

 

[Matterwave]

It would take the bowling ball longer. The other planet will pull the Earth towards it, while the bowling ball's effect on the Earth would be negligible. It will take the bowling ball a factor of sqrt(2) longer than the planet.

See: http://en.wikipedia.org/wiki/Free-fall_time

 

[Drakkith]

The bowling ball accelerates towards the Earth at about 9.8 m/s². The Earth accelerates towards the bowling ball at... well, for our discussion "virtually zero" is accurate enough.

A planet similar to Earth would accelerate towards the Earth at about 9.8 m/s² and the Earth would begin to accelerate towards it by about 9.8 m/s².

 

[Nugatory]

The bowling ball accelerates towards the Earth at about 9.8 m/s². The Earth accelerates towards the bowling ball at... well, for our discussion "virtually zero" is accurate enough.

A planet similar to Earth would accelerate towards the Earth at about 9.8 m/s² and the Earth would begin to accelerate towards it by about 9.8 m/s².

OP specified an original separation of about 25 Earth radius, so the gravitational accelerations will be appreciably less than 1g for most of the travel... but this is more of an additional observation than a correction here. Drakkith's key point, that the Earth and the ball/planet move towards their mutual center of gravity, holds.

 

[Robert Harris]

I appreciate the quick responses, thanks.

I have been debating this issue with a guy who is better educated in physics than I am. He was telling me about Newton's third law of gravitation and that a 100 pound object would fall to Earth at exactly the same speed as a 1 pound object.

I said yes, but in theory, the heavier object should fall a tiny bit faster since it exerted more gravitational pull upon the Earth than the smaller one. Of course that difference would be incredibly small and immeasurable in Newton's time.

I then said, what if we replace those two objects with an entire planet and a bowling ball, and asked him the same question that I posed to the forum.

His response was that the bowling ball would compensate for the greater gravitational pull of the planets, by accelerating faster, and that it would strike the Earth in the same amount of time as the planet.

Any thoughts on that, pro or con?

 

[Andrew Mason]

I appreciate the quick responses, thanks.

I have been debating this issue with a guy who is better educated in physics than I am. He was telling me about Newton's third law of gravitation and that a 100 pound object would fall to Earth at exactly the same speed as a 1 pound object.

I said yes, but in theory, the heavier object should fall a tiny bit faster since it exerted more gravitational pull upon the Earth than the smaller one. Of course that difference would be incredibly small and immeasurable in Newton's time.

I then said, what if we replace those two objects with an entire planet and a bowling ball, and asked him the same question that I posed to the forum.

His response was that the bowling ball would compensate for the greater gravitational pull of the planets, by accelerating faster, and that it would strike the Earth in the same amount of time as the planet.

Any thoughts on that, pro or con?

This is just a matter of applying the Law of Universal Gravitation. The forces of each of the two planets (earth and planet x) on the other would be:

$\vec{F} = -\frac{GM_eM_x}{R^2}\hat{R}$ where $\hat{R}$ is the unit radial vector from the centre of the earth/planet x and R is the distance between the centres of the two bodies.

The accelerations are, initially, the same. However, because the Earth would move toward the planet x and vice versa, the force/acceleration would increase more rapidly than in the bowling ball case. Also, because planet x is bigger than the bowling ball, the separation between their surfaces is less so there is less initial distance to cover in the "fall".

AM

 

[Pythagorean]

Newton's third law is appropriate here:

$$F_{21} = -F_{12}$$

and Newton's second:

$$F = ma$$

which means:

$$m_2 a_2 = -m_1 a_1$$

then, solving that for $a_2$ and substituting below, the relative acceleration between the two bodies would be:

$$a_2 -a_1 = (1+\frac{m_1}{m_2})a_1$$

If $m_1 \ll m_2$, as in the bowling ball case, then the relative acceleration is just the acceleration of the bowling ball, $a_1 = G\frac{m_2}{r^2}$ where $m_2$ is the mass of Earth. $a_2$ is essentially 0.

Now... with two planets, $m_2$ is still the same, so $a_1$ is initially still the same, but in our coupling term above, you now can't neglect the additional $\frac{m_1}{m_2}$ term and you can see how their mutual acceleration can be twice as great, for instance, if $m_1=m_2$.

 

[Buckleymanor]

I appreciate the quick responses, thanks.

I have been debating this issue with a guy who is better educated in physics than I am. He was telling me about Newton's third law of gravitation and that a 100 pound object would fall to Earth at exactly the same speed as a 1 pound object.

I said yes, but in theory, the heavier object should fall a tiny bit faster since it exerted more gravitational pull upon the Earth than the smaller one. Of course that difference would be incredibly small and immeasurable in Newton's time.

I then said, what if we replace those two objects with an entire planet and a bowling ball, and asked him the same question that I posed to the forum.

His response was that the bowling ball would compensate for the greater gravitational pull of the planets, by accelerating faster, and that it would strike the Earth in the same amount of time as the planet.

Any thoughts on that, pro or con?

Your friend is correct for most cases of object's falling.There won't be a measurable amount of difference between objects of 1 pound and 100 pounds.
What is different is when large bodies fall. When two Earth sized planets attract each other they both fall at the same rate towards each other, therefore there closing speed is greater, than say a bowling ball falling towards one of them.

The bowling ball attracts the earth, and the Earth attracts the bowling ball but the attraction of the bowling ball on the earth, is no way as strong as another Earth sized planet.

The two Earth sized objects only have to travell half the distance before the collide so they take less time.

 

[rcgldr]

This depends on if you define "fall" as the change in speed of the object relative to it's initial state, in which case the size of the object doesn't matter, or if you define "fall" as the change in closure rate between the object and earth, in which case the size matters. The time to impact will be a function of closure rate.

 

[Robert Harris]

Thanks again for the detailed replies. But the crux of the argument with my friend is my original claim that a 100 lb object would fall faster than a 1 lb object, for the same reason that a planet would.

Is there a dividing line somewhere? Would the moon fall faster than the bowling ball?

How about a one mile wide asteroid?

It seems to me, in my infinite ignorance, that the same rule should apply in all cases.

 

[Pythagorean]

Thanks again for the detailed replies. But the crux of the argument with my friend is my original claim that a 100 lb object would fall faster than a 1 lb object, for the same reason that a planet would.

Is there a dividing line somewhere? Would the moon fall faster than the bowling ball?

How about a one mile wide asteroid?

It seems to me, in my infinite ignorance, that the same rule should apply in all cases.

I think if you look at my analysis you can see that yes, the bigger the object (m1) for a fixed m2 (earth), the greater the acceleration.

 

[Buckleymanor]

The bowling ball accelerates towards the Earth at about 9.8 m/s². The Earth accelerates towards the bowling ball at... well, for our discussion "virtually zero" is accurate enough.

A planet similar to Earth would accelerate towards the Earth at about 9.8 m/s² and the Earth would begin to accelerate towards it by about 9.8 m/s².

Would the two planets accelerate towards each other at 9.8 m/s2 as they have twice the mass of one Earth combined.

 

[Robert Harris]

ModusPwnd, I understand (sort of) what Newton is saying and that a feather would fall to the ground at the same, measurable speed as a hammer.

What I am suggesting is that since the hammer is pulling slightly harder on the Earth than the feather, that it would fall slightly faster - the difference of course, being too small to measure.

If that is correct, then the difference becomes much greater when we consider very large objects, like moons, asteroids, planets, etc.

It seems that these various objects have one thing in common. They are each pulling on the Earth and the Earth is pulling on them. Therefore, shouldn't the same rule apply in each case?

If so, then if a planet will collide with the Earth faster than a bowling ball, a 100 lb object should collide faster than a 1 lb object.

 

[Pythagorean]

What I am suggesting is that since the hammer is pulling slightly harder on the Earth than the feather, that it would fall slightly faster - the difference of course, being too small to measure.

That's theoretically correct. The feather pulls on Earth. The hammer pulls on Earth more.

 

[Andrew Mason]

Newton's second law is appropriate here:

$$F_{21} = -F{12}$$

and Newton's First:

$$F = ma$$

which means:

$$m_2 a_2 = -m_1 a_1$$

I think you meant to say:

Newton's third law is appropriate here:

$$F_{21} = -F{12}$$

and Newton's Second:

$$F = ma$$ ...

I think if you look at my analysis you can see that yes, the bigger the object (m1) for a fixed m2 (earth), the greater the acceleration.

? You are confusing us! The force is greater but the acceleration is the same relative to an inertial point.

 

[Pythagorean]

Oops, yeah, wrong numbers on Newton's

? You are confusing us! The force is greater but the acceleration is the same relative to an inertial point.

But we're talking about the relative acceleration. I explicitly took the difference between a1 and a2 as such. Robert Harris is considering the motion of the Earth, too. So it is indeed larger for a larger m1, given that m2 is fixed (Earth).

Remember the OP's question:

Would it take the bowling ball longer to collide with the Earth than the other planet, the same amount of time, or less time?

 

[Andrew Mason]

If so, then if a planet will collide with the Earth faster than a bowling ball, a 100 lb object should collide faster than a 1 lb object.

If you raise a kilogram mass by one metre, the centre of the Earth has to move away from its initial position by 1/.594e25 = 1.7e-25 m.

To give you an idea of how much this is, a hydrogen atom is about 10e-10m in diameter and a proton diameter is about 10e-15 m (femtometre). So a kg. mass falling 1 m. is going to move the Earth about one ten-billionth of the diameter of a proton.

Note: If you happen to drop a hammer and a feather at the same time, they take identical times to fall.

Also: if you drop a hammer at the same time that someone else is doing the same thing on the diametrically opposite side of the earth, the earth, even in theory, does not move at all.

Since there are things being raised and lowered all the time around the earth, whether the feather or hammer takes a longer time to drop, when dropped separately, really depends on what else is happening around the Earth when you do the experiment.

AM

 

[Pythagorean]

If you happen to drop a hammer and a feather at the same time, they take identical times to fall.

As rcgldr pointed out, it depends on how you define fall. I get the impression OP and I are defining fall differently than you (relative distance vs. inertial reference frame).

 

[Andrew Mason]

As rcgldr pointed out, it depends on how you define fall. I get the impression OP and I are defining fall differently than you (relative distance vs. inertial reference frame).

If they are dropped at the same time from the same height why would one reach the surface of the Earth before the other? The Earth cannot accelerate differently for each if they are dropped at the same time.

AM

 

[Pythagorean]

Oh, I see what you mean. In both definitions, they would contact Earth at the same time under those initial conditions, despite giving different values for the total acceleration. I agree.

 

[Andrew Mason]

Oh, I see what you mean. In both definitions, they would contact Earth at the same time under those initial conditions, despite giving different values for the total acceleration. I agree.

I don't quite understand why they would give different values for either the total acceleration or the relative acceleration.

AM

 

[Ott Rovgeisha]

Hello, forum members.

I am the mysterious person that Robert had the argument with.
Let me explain what the argument was actually about.

Basically what I said, is this:

When there is a force between a ball and the earth, then that force is MUTUAL for both bodies.
Robert seems to be very uncomfortable with the idea that a tiny ball pulls on the Earth with the same amount of force that the Earth pulls on the ball.

But that is exactly what happens. There is ONE MUTUAL force between the EARTH and the BALL, that force is acting on BOTH of them, but it is ONE SINGLE FORCE.

I think the most correct way to put is this: there is ALWAYS only one force between two bodies that attract each other or repel each other; this one force acts on BOTH bodies; this is why we can write the F= ma. The F being the action upon either bodies.

Earth is NOT pulled LESS by the ball. The Earth is pulled EXACTLY as strongly as the ball is pulled.

Robert, forgive me, if I am misrepresenting you, but basically, Robert does not like that idea.
According to him, the EARTH does not accelerate towards the ball, because the ball pulls on it with a force that is much weaker than the force that Earth exerts of the ball.

But that is a misconception. The REAL reason, according to Newton, is that the MUTUAL FORCE between the BALL and the EARTH just isn't strong enough to accelerate the Earth upwards in a meaningful way, because the mass of the Earth is huge. It is however enough to accelerate the ball visually. This is the reason why the ball accelerates towards Earth not vice versa.

To prove this "juridically" I used the very famous law of gravity (simplified version):

FORCE = (mass1*mass2)/distance squared.

Note that there is ONLY ONE FORCE: one mutual pull! Not two! Both of the masses are needed to create this force! Yes there are two ACTIONS of the force (the vector quantities), but only one magnitude of the force!

And reading Newton's "Principia", which I am sure, by FAR not every physicist has, there are these quotes:

"The mutual actions of two bodies upon each other are always equal, and directed to contrary parts."
"Whatever draws or presses another, is always as much pressed or drawn by that other"

These are only two of many quotes relevant to this issue.

This is why Newton's idea was ingenious: he dared to say that a tiny pea would pull on the Earth exactly as strongly as the Earth pulls on the pea. This is non-intuitive, but it makes sense.

 

[Andrew Mason]

Hello, forum members.

Basically what I said, is this:

When there is a force between a ball and the earth, then that force is MUTUAL for both bodies.
Robert seems to be very uncomfortable with the idea that a tiny ball pulls on the Earth with the same amount of force that the Earth pulls on the ball.

But that is exactly what happens. There is ONE MUTUAL force between the EARTH and the BALL, that force is acting on BOTH of them, but it is ONE SINGLE FORCE.

You seem to be redefining "force". The forces on each have the same magnitude. But the Earth and the ball have two different accelerations. Since F = ma, there are, by definition, two forces. They both have the same physical cause, but there are two forces.

Robert, forgive me, if I am misrepresenting you, but basically, Robert does not like that idea. According to him, the EARTH does not accelerate towards the ball, because the ball pulls on it with a force that is much weaker than the force that Earth exerts of the ball.

I don't think he is saying that. He appears to be saying that the 100kg mass exerts more force on the Earth than the 1 kg ball. So, he concludes, the Earth accelerates at a greater rate toward the 100kg mass than toward the 1 kg. ball.

This is why Newton's idea was ingenious: he dared to say that a tiny pea would pull on the Earth exactly as strongly as the Earth pulls on the pea. This is non-intuitive, but it makes sense.

This is correct. But I don't think Robert is disagreeing with you on this point. He is just saying the 100kg ball pulls on the Earth with more force than the pea does, so the acceleration of the Earth toward the ball is greater than the acceleration of the Earth toward the pea. And that is quite true.

Welcome to PF, by the way!

AM

 

[Ott Rovgeisha]

Thank you for welcoming me, Andrew!

That is interesting. If Robert says that the 100 kg ball pulls on the Earth with more force than the pea does, so the acceleration of the Earth toward the ball is greater than the acceleration of the Earth toward the pea, then we do not disagree at all :D.

As for two forces vs one... I have always thought of F=ma as F being the action upon a body, force being the cause. The mutual force between two bodies causes these two actions on both bodies. Newton himself described it this way. He is talking about mutual draw (attraction) or press (repulsion) that acts on both bodies. I sometimes teach my students this way, because it reduces the risk of the misconception, that force is an intrinsic property of a body, which it is not.

But I think that is not too relevant here. It does not change the more important thing: the fact that the pea pulls on the Earth EXACTLY as strongly as the Earth pulls on the pea.

It is not correct to say this:

"The Earth does not accelerate towards the pea, because the pea pulls on the Earth with a smaller force than the Earth pulls on the pea."

This is not correct!
The correct way of looking at this is this way:
"The Earth hardly accelerates towards the pea, because mutual attraction between the pea and the Earth is not big enough to make the massive Earth accelerate towards the pea. It is however, big enough, to make the pea accelerate."


Another thing that is of course very non intuitive is the fact that even though the bowling ball is pulled down with a force that is some 1000 times greater than the force that pulls the pea, they STILL have the same acceleration! This is because 1000 times more massive bowling ball needs to be pulled by 1000 times larger force in order to have the same acceleration as the pea, which has a 1000 times smaller mass.

 

[Pythagorean]

I don't quite understand why they would give different values for either the total acceleration or the relative acceleration.

AM

In the case of the relative frame, the Earth's motion towards the feather and hammer due to their combined gravitational pull on Earth would add to the relative acceleration. An observer in an inertial reference frame would see a slightly smaller acceleration for the feather/hammer.

 

[Andrew Mason]

Another thing that is of course very non intuitive is the fact that even though the bowling ball is pulled down with a force that is some 1000 times greater than the force that pulls the pea, they STILL have the same acceleration! This is because 1000 times more massive bowling ball needs to be pulled by 1000 times larger force in order to have the same acceleration as the pea, which has a 1000 times smaller mass.

It seems that you and Robert agree on this. It is just that Robert would point out that since the force of the gravity is 1000 times greater with the ball, the Earth acceleration toward the ball is 1000 times greater than its acceleration toward the pea. He acknowledges that the accelerations of the Earth are immeasureably small in both cases.

AM