There is no gravity. The Earth sucks. 
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Smit
"Be excellent to each other."
Bill & Ted -
Considering extreme situations is a good way to expose the simplifying assumptions made in saying the dropped objects would strike the ground at the same time. But I'd like to return to the original proposition of dropping the ping pong ball and the bowling ball at the same time, in a perfect vacuum, and argue that the ping pong ball will strike the ground first!
The law of universal gravitation says the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. And the second law of motion (F=ma) says the acceleration of a given mass is inversely proportional to the force applied. That is, larger masses experience more gravitational force, but also require more force for the same acceleration. Therefore the acceleration due to the force of gravity is independent of the masses for a given distance between the centers of gravity.
The earth does experience a force toward the balls, and therefore an acceleration toward them, but if both balls are dropped at the same time we can ignore that acceleration, not because it is infinitesimal, but because the earth accelerates toward both balls. And we can ignore the attraction between the balls (three body problem) because it is (almost) perpendicular to the direction of fall toward the earth that is being timed. The balls will hit the earth a little closer to each other than when they were dropped, but their acceleration toward the earth is not affected by their mutual attraction to each other.
Now, if the two balls are to fall the same distance before hitting the earth, their bottom surfaces must start from the same height. But the larger diameter of the bowling ball, compared to the ping pong ball, means that its center of mass starts at a larger distance from the center of mass of the earth. It's a difference of a few inches in about 4000 miles, but the inverse relationship between gravitational force and the square of the distance means that the bowling ball's acceleration towards the earth will be less than that of the ping pong ball, and the ping pong ball will hit the earth first!
Any more nits to be picked?
Sid
(Realize that it was a long time ago, but I got a masters in physics, did all the course work for a Ph.D., and worked about a year in an atomic accelerator lab studying one energy state of the helium negative ion for my thesis (really!). Analyzing the data was my first exposure to computer programming (loooong time ago). I decided that was a whole lot more fun than what I was doing, quit the thesis work, and never looked back.)Comment
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Sid I have to admit that makes some sense...
What's all the "three body probem" stuff? Is this really unsolvable? I really did think this was a simple question at first...
Also, I could be wrong, but I don't really think that you'd explode in a vacuum. Maybe you'd look like you gained a few pounds, but I can't see actually coming apart.
radhak, your comment about "'letting it go' would be as tough as throwing the earth to the moon" is hilarious -- never even crossed my mind. I have this funny image of a bowling ball stuck to my hand -- actually it would feel like I was doing a one-armed handstand on the bowling ball!
All that aside, though, there are a lot of assumptions involved because it is such as "extremem situation" as Sid puts it.
I guess what it comes down to is this: Considering all other things are equal -- including size and shape of the balls even -- and you have one ball that's much much more massive than the other. Drop them at different times, even, to avoid the "three body" issue. Won't the more massive one hit the ground faster -- even if it's just a tiny bit faster?online at http://www.theFrankes.com
while ( !( succeed = try() ) ) ;
"Life is short, Art long, Occasion sudden and dangerous, Experience deceitful, and Judgment difficult." -HippocratesComment
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With separate drops and equal diameter balls I agree with you. My argument relied on a simultaneous drop of the balls to ignore the acceleration of the earth toward the balls, which would be greater for the more massive ball; and on unequal diameters, which increases the distance between centers of gravity for the larger ball.
The general three body problem has not been solved with the analytic methods which derive elliptical orbits for two bodies. But numerical techniques with enough computer power can calculate the subsequent motions of three bodies for a given set of initial conditions (locations and velocities) to whatever degree of accuracy you need.
SidComment
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I have no idea how to even approach the problem. I do know that if you posed the question to my BIL he'd ram the two objects through this super-conducting super-colider and quantify the Higgs particle content of each object.
JRComment
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One-armed handstand? I hope you have VERY strong arms. Instead of being about 6.4 million meters from the center of mass you would be about 0.15 meters from the center... So you'd be feeling something like 17 quadrillion gravities where you're touching the ball and and a paltry 86 trillion gravities at your feet. This would be bad
Note: My cheapie calculator doesn't have that many digits so, if the numbers are slightly off I'll blame the bad math in my head
Kristofor.Comment
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JR, please consider this questions posed to your BIL! I have no idea what you're talking about, but it sure sounds like fun!
Yes, good point -- My arm could handle 15 quadrillion G's maybe, but not 17.One-armed handstand? I hope you have VERY strong arms. Instead of being about 6.4 million meters from the center of mass you would be about 0.15 meters from the center... So you'd be feeling something like 17 quadrillion gravities where you're touching the ball and and a paltry 86 trillion gravities at your feet. This would be bad
Note: My cheapie calculator doesn't have that many digits so, if the numbers are slightly off I'll blame the bad math in my head
Kristofor.
online at http://www.theFrankes.com
while ( !( succeed = try() ) ) ;
"Life is short, Art long, Occasion sudden and dangerous, Experience deceitful, and Judgment difficult." -HippocratesComment
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The pic is from the ATLAS detector at CERN, taken last Friday. They intend to collide protons and photons within a huge electro-magnetic array. The current going through each of the magnets is something like 20,000 amps.
The idea is that if you collide these atoms within the huge magnet, subatomic particles will be given off. These particles live a very short time, nanoseconds. During that time their presence will be deduced by changes in the magnetic field. The Higgs particle is a theoretical entity, smaller than a quark. That's what they're looking for.
Given that my BIL is trying to unearth the basic elements of all creation, I don't think I'll be posing any bowling-ball/ping pong-ball queries!
JRJRComment
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If your holding both the bowling ball and the pingpong ball and let go I would think that the pingpong ball would attach to the bowling ball well before they get near the ground. The Pingpong ball and bowling ball would only be about 6'-7' apart, maybe 7.5' compared to 180'+. That pingpong ball would snap to the bowling ball as soon as you let go.Ric
Plan for the worst, hope for the best!Comment
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What about this?
In the original question, the bowling ball had the mass of a planet. If we stretched our arms out, we could maybe get them 6-8 feet apart, which is significantly smaller than the 180+ feet tall tower. Therefore, the bowling ball and ping pong ball would meet before either hit the ground.
Assuming the bottom of the balls were aligned before they were dropped, we can look at two situations. The first (shown in figure 1) shows the bottoms aligned and they are touching. The straight line is connecting the centers, showing the line along which they would be attracted toward each other. In the second (which is shown in figures 2 and 3) they are not initially touching. The red ball shows the ping pong ball moving to the point it touches the bowling ball. The mass of the bowling ball would pull the ping pong ball, causing it to slow so the bottoms would no longer be equal (the lowest point of the bowling ball would be lower). The greater the distance between the two balls, the greater the difference in their lowest point.
I would think that if the two were initially touching, they would hit at the same time, and if they were separated by a distance, the bowling ball would hit first (it's lowest point would be lower than that of the ping pong ball's).
Does this make sense, or what am I missing?
TomComment
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I think the ping pong ball would hit first because the gravity of the two massive objects (earth and bowling ball) would pull it in between them It would actually levitate.
and if you had a bowling ball the mass of the earth, I'd be concerned about the effects on the rotation of the earth first.AlexComment
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In my first post, when I said "return to the original proposition," I meant before the introduction of the ultra-massive bowling ball. My whole argument was for standard ping pong and bowling balls. Sorry if that wasn't clear.
Sid
P.S. Note that I referred to the acceleration of the earth toward the balls as "infinitesimal" because I was assuming a standard bowling ball, not a compressed planet.Comment
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They would fall at the same speed. I figure the ping-pong ball would fall toward the other (theoretical) ball first because it's only a metre away, and the two would fall as one the 56-odd metres to the ground from there.
That's what would happen in Canadian dollars at least.AndrewComment
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If anyone cares to actually calculate it, you use Newton's law of gravitation
force of gravity between two objects:
Fg=(Gm1m2)/(r^2)
G=6.67 x 10^-11 [(Nm^2)⁄(kg^2 )]<---the part in between [ ] is just units
m1=mass of earth in kilograms
m2=mass of bowling ball (earth) or ping pong ball in kilograms
r=distance between objects in meters
your answer will be in Newtons (N)
without doing the calculation but thinking it through: (this has already been eluded to in previous posts)
if they were dropped at separate times so that the ping pong ball wasn't attracted to the bowling ball, also ignoring that there is gravitational pull between the earth and other planets;
the bowling ball and earth would accelerate toward each other equally as fast.
the bowling ball would meet the earth in less time than the ping pong ball would.
reasoning:
the ping pong ball's mass (and all other "moveable" masses) on earth is insignificant compared to the earth's mass so it has an insignificant effect on the earth but the earth has a significant effect on the ping pong ball. basically the earth sits still and the ping pong ball moves to it. they always fall accelerating (ignoring air resistance) at at 9.8 (m/s^2) this is why hammers and feathers hit the ground at the same time in a vacuum.
when the bowling ball's mass is equal to the earth then they have an equal and significant amount of effect on each other so they accelerate towards each other with equal force and equal velocity. that closing speed would be greater than the 9.8 (m/s^2) that we are accustomed to as earth dwellers.
or
think about earth/bowling ball system as 2 large identical magnets attracted toward each other.
think about earth/ping pong ball system as 1 extremely large magnet and one tiny magnet attracted toward each other.
if both balls were dropped at the same time the ping pong ball would be most attracted to the mass closest to it. if they were equidistant then potentially the ping pong ball would hit earth before the bowling ball but only because it was pulled between the two objects (with equal distance between them) and then smashed between them as they collided. so it would it would hit both the earth and the bowling ball at the same time.Opportunity is missed by most people because it is dressed in overalls and looks like work. - Thomas EdisonComment
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Alex, the mass is not supposed to matter because there are several assumptions to this particular model. One is that the mass of the Earth is so huge compared to the objects falling, and that the diameter of the Earth is very very large compared to the distance through which the objects are falling.
Because of these two assumptions, the acceleration due to gravity, g, near the Earth's surface is deemed to be constant. Strictly speaking, it's not because it will be dependent on the distance between the center of mass of the object and that of the Earth. That's why g is different when you're on top of Mount Everest and when you're down by the beach. But for small heights/altitudes, like the Tower of Pisa, the g is practically the same near the bottom of the tower and near the top. (For engineers like myself, that's really convenient.)
An object can experience a gravitational force by the Earth. But at the very same time, that object also exerts a gravitational force of equal magnitude on the Earth. F = ma, and even though the magnitude of the forces F are equal for both the object and the Earth, the masses are hugely different so that the accelerations are also vastly different (in inverse proportion). For example, if the mass of the object is 1/1000 of the mass of Earth, then the acceleration of Earth is 1/1000 that of the object.
If you have an object with the mass of Earth close to Earth, and they are both free to move, then they will both move with the same accelerations towards each other every time. Someone mentioned the Earth jumping to meet the massive object, and he's technically correct. That's assuming they're starting from rest and are not moving around each other like the Earth and the Moon or the Earth and the Sun.
If we isolate the object (like your ball), its acceleration at any point measured from the center of the Earth doesn't matter on how massive the object is. All the acceleration will depend on is the mass of the Earth and the distance from the center of the Earth. If the Earth is tethered so that it can't move, then an object falling from the Tower of Pisa will take the same time to reach the bottom no matter how massive it is.
If you're dropping the two identical-sized balls at the same time, and one of them is as massive as the Earth, then the second ball will experience a horizontal gravitational force towards the other ball and a gravitational force downwards towards the Earth. Let's say the object with the same mass as the Earth has a diameter that is 1/1000 that of the Earth's diameter. Since gravitational force varies inversely with the square of the distance, the acceleration due to gravity at the surface of the smaller ball will be 1,000,000 times g at the surface of the Earth. (Someone correct my math if I'm wrong.)
In any case, that's just a detail. The less massive ball will initially move in a projectile motion towards the other ball and towards the Earth until it basically sticks to the massive ball (time it takes depends on how far away they were from each other to begin with) and they both move down with the same speed and acceleration towards the Earth.
Also, this is not a classical three body problem. And relativity becomes important when speeds exceed about 10% of the speed of light. I don't think it happens in this case.
Now I've discovered it takes about two days before I can't shut up anymore. I'm sure I'll do a better job staying away from the next physics question.
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