## Objects Moving Apart Under Gravity?

Let's say that you shoot two balls straight up into the air (in vacuum) from the surface of the Earth side by side with one at a higher velocity and the other at a lower velocity.

The higher velocity ball will go higher and land later on - of course - than the lower velocity ball. And of course the distance between them will increase.

Will the distance between those balls accelerate as they move further from the Earth?
Will the distance between those balls continue to accelerate as the lower velocity ball begins to fall back to Earth?
Will the distance between those balls continue to accelerate as the higher velocity ball begins to fall back to Earth?

My overall question is do the balls appear to accelerate away from each other throughout the entire trajectory?
 Recognitions: Gold Member ]Will the distance between those balls accelerate as they move further from the Earth? Will the distance between those balls continue to accelerate as the lower velocity ball begins to fall back to Earth? Will the distance between those balls continue to accelerate as the higher velocity ball begins to fall back to Earth? My overall question is do the balls appear to accelerate away from each other throughout the entire trajectory? You best way to answer these questions is to write equations for each ball's motion and compare them. Gravity will accelerate each ball at a steady 32 ft/sec/sec towards the earth...that steady rate of acceleration slows the balls from their initial velocity, stops each ball at a maximum altitude, then speeds the balls up as the return to earth... so the general answer is they accelerate steadily due to gravity...regardless of their individual masses...weights!!! v = v' - gt ; v^2 = v'^2 - 2gy ; y = v't - 1/2gt^2.... etc where v' is the initial vertical velocity.
 My question does actually relate to Cosmology but fair enough maybe someone in homework will help me. Thanks Naty1. I have tried to find example plotted trajectories of two objects with different velocities by searching the net but the only examples I keep finding are example plotted trajectories of two objects with the same velocity (at different angles). I need the first and not the later. It pertains to a further question I have namely how would we know if the Universe were in the process of crunching when it would still appear to us to be expanding?

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## Objects Moving Apart Under Gravity?

Will the distance between those balls accelerate as they move further from the Earth?
No, but the distance increases constantly.

Will the distance between those balls continue to accelerate as the lower velocity ball begins to fall back to Earth?
No, but the distance will continue to increase constantly.

Will the distance between those balls continue to accelerate as the higher velocity ball begins to fall back to Earth?
No, but the distance will continue to increase constantly.

My overall question is do the balls appear to accelerate away from each other throughout the entire trajectory?
No, but the distance will continue to increase constantly.
This can be expressed as ball 1 has a constant velocity relative to ball 2 throughout the entire trajectory.

Let's say the ball 1 has an initial velocity of 60 m/s and ball 2 has an initial velocity of 50 m/s then ball 1 will always have a velocity of 10 m/s relative to ball 2.

You can work this out using the equation given by Naty1 for the height (y) of each ball at any given time t and subtracting the results from each other at any given time t so:

ydiff = y1-y2 = (v1*t -1/2gt^2) -(v2*t-1/2gt^2 ) = (v1-v2)t -1/2gt^2+1/2gt^2 = (v1-v2)t

ydiff is directly proportional to t since (v1-v2) is a constant (initial velocities) and the plot of ydiff versus time is imply a straight line.
 Blog Entries: 6 This is the plot of the equations I gave in the last post. The blue curve is the trajectory of ball 1 with an initial velocity of 60 m/s and the red curve is the trajectory of ball 2 with an initial velocity of 50 m/s. The diagonal green line is the plot of the difference in height at any time t. Attached Thumbnails

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 Quote by gonegahgah It pertains to a further question I have namely how would we know if the Universe were in the process of crunching when it would still appear to us to be expanding?
First, you should realise that the case of the universe you would have to plot the gravitational accelertion of ball one to ball 2 because there is a not a large central source of gravity represented in your example by the Earth in cosmology. Second you would need to use the equations of General Relativity and not those of newton in cosmolgy. The closest solution is probably represented by something based on the interior Schwarzschild solution or maybe even better by the FRW metric because that allows for the motion of all the expanding/contracting falling objects that are in turn the source of gravitation too. It is far from a simple question. From a personal point of view (just my gut feeling) you are probably right on a deeper level that an expanding universe will look very much like a contracting universe bu that is no simple matter to prove.

All the best ;)
 Hi Kev, thanks for your answer. I would prefer an actual plot please if you could... In my head I would expect the object with higher initial velocity to always be higher than the object with lower initial velocity. The lower object would therefore be in higher gravity and would therefore accelerate quicker back to Earth than the higher object. In this respect the lower initial velocity ball would move back to Earth with greater acceleration throughout its path than would the higher initial velocity ball. In this respect the lower ball's speed would change a greater amount in the same amount of time than the higher ball's speed. In this respect they would appear to be accelerating away from each other at all times. Am I correct to say that your formula is showing g to be the same for each of the balls at the same time when it is not? Can anyone provide me with a plot to confirm or negate this please? Oops! Just saw you attached picture Kev. Have to wait for it to be approved to view it...

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 Quote by gonegahgah Hi Kev, thanks for your answer. I would prefer an actual plot please if you could... In my head I would expect the object with higher initial velocity to always be higher than the object with lower initial velocity. The lower object would therefore be in higher gravity and would therefore accelerate quicker back to Earth than the higher object. In this respect the lower initial velocity ball would move back to Earth with greater acceleration throughout its path than would the higher initial velocity ball. In this respect the lower ball's speed would change a greater amount in the same amount of time than the higher ball's speed. In this respect they would appear to be accelerating away from each other at all times. Am I correct to say that your formula is showing g to be the same for each of the balls at the same time when it is not? Can anyone provide me with a plot to confirm or negate this please?
OK, the equations mentioned by Naty1 and used by me are the regular Newtonian ones that assume g is constant and the height above the ground is much smaller than the radius of the Earth. The calculations for varying G are much more complicated but you are right that in general the acceleration for the slower particle at a lower altitude will be greater than the accelration on the faster higher particle at any given time.

On the other hand, you would have to show there is a large central source of gravity for the universe that acts like the Earth in your example.

I think this is far as we can go with this in the homework section. I think you will have to rephrase your question and resubmit it to the Cosmology forum in a manner that makes it clear that you are referring to cosmology.
 Thanks for your answers Kev. What are supposed to be the big drivers of a crunch? Is it gravity? If I wait will the kind moderators move it back to the cosmology section?

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 Quote by gonegahgah Thanks for your answers Kev. What are supposed to be the big drivers of a crunch? Is it gravity?
Yes, good ol gravity. But you need more gravity than there is dark energy which is a sort of negative gravity accelerating the expansion of the universe. The consensus of the Cosmolgy experts at the moment is that the dark energy greatly exceeds the the mutual gravity of galaxies and that the universe will never collapse but will expand at an ever increasing rate until no galaxies or stars are visible from any given location. That is the "big rip" rather the the "bug crunch". There is a minority that think dark energy is just a misinterpretation of the observations but that is definately not the "accepted mainstream view", (at this point of time). Then again, the point of view that Sun does not orbit around the Earth was not the mainstream view back in Galileo's time ;)

The trouble with dark energy is that it requires an extremely fine balance. If the dark energy was initialy some parts per million greater than it is , then the the big rip will already of happened. Some think the extraordinary coincidence required to explain the observations is "too much".
 Thanks Kev. Even without dark matter I would like to think there is something like an escape velocity which once it is initiated then things will never fall together again under gravity ever. It works for planets doesn't it? If that is so I don't see why it wouldn't work for a universe spreading apart. That's speculation on my part of course and I don't know how that even remotely fits in with any of today's positions on expansion. Also from what I understand of it; the universe acts as its own centre of gravity. Assuming there is a centre somewhere - though non-identifiable from within - then there will always be more matter on your side towards, through and beyond the centre then there is on your opposite side away from the centre. So this would then always act as the centre of gravity towards which you are most strongest attracted (ie. there is more stuff towards the inside than towards the outside). That is assuming a couple of things: that there is a centre somewhere (even if it is unidentifiable as such from our perspective), that there is an edge somewhere, and that the expansion occurs througout the universe from the edges through the centre in a common way.
 I can see that plot of the two trajectories now Kev. Thanks for that. It is good because even with this it is possible to see that for each slot of time the two balls constantly expand apart even during their transition and crunch phases. So 'if' this represented the behaviour our universe then we could not know if we were in a phase of expansion or a phase of contraction by observation.
 I would like to try and answer this question by simply visualizing the experiment using 'common sense' if you will: Ball 1 is moving faster going up = the distance between them increases. The slower ball 2 stops going up while the faster ball 1 continues upward a little longer = the distance between them increases further. Ball 1 stops going upward while ball 2 has already started accelerating downward = the distance between them increases further. Ball 1 starts accelerating downward while Ball 2 continues to accelerate downward = the distance between them increases further. Ball 2 hits the ground while Ball 1 continues accelerating downward = the distance between them decreases until Ball 1 hits the ground, finishing the experiment. Conclusion: The distance between them increases until the "slower" ball hits the ground. I can see no reason for the distance between them to "accelerate" or, in other words, grow exponentially, throughout the entirety of the "growing" stages. In fact I think that's impossible in this experiment although I'm not sure how to articulate why.
 I really would like to understand the calculations involved here and I hope you all will indulge me before closing this discussion. And I am NOT a student, just an inquisitive Idjot: Suppose each ball weighs 10 lbs. Suppose Ball 1 is hit upward with 50 lbs of force and Ball 2 is hit upward with 40 lbs of force. Here are my questions: 1: What will be the max speed of each ball on the way up? 2: What will be the max height of each ball? 3: At what height will each ball reach it's maximum speed? 4: At what rate will each ball slow down? 5: I can understand the balls appearing to accelerate away from one another on the way down, but not on the way up. It seems to me that they would appear to be moving away from each other steadily on the way up, or at least for part of it. My so called 'common sense' tells me that the relationship of the 2 balls couldn't possibly be exactly the same fighting against gravity as submitting to it, and also during separate transitions in states. Being that G is a set number, shouldn't that mean that the difference in the balls initial velocity could alter the results of said acceleration in distance. In other words, even if they really do appear to accelerate away from one another the whole time, wouldn't that rate have to be different for each stage of the experiment due to the initial difference in velocity against G? 6: Do I sound insane?
 Hi Idjot As soon as the balls leave the 'ground' they begin to decelerate. So their fastest speed is at take off and corresponds to their speed at landing. In the plot provided by Kev the deceleration (g) remains constant throughout the path of the balls (whereas normally g decreases with height). This means that the speed (not the distance travelled) of each ball will be altered by the same amount during each time period. This is because acceleration acts on speed. Here is an example showing the height of the balls after time 1, 2, 3, 4, 5. Ball 1 is shown as a) and its speed is shown as s1 and its height is shown as y1. Ball 2 is shown as b) and its speed is shown as s2 and its height is shown as y2. The height is shown as previous height plus current change (note that current change is half the speed): a) s1=6,y1=0+3; s1=4,y1=3+2; s1=2,y1=5+1; s1=-0,y1=6+0; s1=-2,y1=6-1. b) s2=4,y2=0+2; s1=2,y2=2+1; s1=0,y2=3+0; s2=-2,y2=3-1; s2=-4,y2=2-2. So for each time slot you get y1,y2 as: t1=3,2; t2=5,3; t3=6,3; t4=6,2; t5=5,0 The distance between the two for each time interval goes as: 1, 2, 3, 4, 5 So you can see through this example, and also by the plot provided by Kev, that the change grows by the same amount whether both balls are still climbing, the lower only is dropping or when both are now dropping. I hope this clarifies it for you idjot. If it doesn't let us know and I, Kev, or someone will try to explain it better.

 Quote by gonegahgah Hi Idjot The distance between the two for each time interval goes as: 1, 2, 3, 4, 5
First let me say thank you for your willingness to help me understand this in it's simplest form.

Based on what you've layed out here let's assume that the time interval between observations is 1 second and the distance measurements are in feet.

That would mean that the balls do not appear to be accelerating away from one another at all but rather they appear to be moving away from one another inertially at 1/2 foot per second each (or one of them moving away at 1 foot per second while the other sits).

No acceleration, right? Just a steady increase in the distance between them, right?

What I'd like to know now is why they would both decelerate at the same rate when one has more momentum.

 Quote by Idjot No acceleration, right? Just a steady increase in the distance between them, right?
You are correct. Our example has the deceleration force as constant so the two balls would move away from each other at constant speed; as shown.

This is not a real world model though. Under real world conditions the higher ball will feel increasingly less gravity at all times than the lower ball as it gets further apart. So it will experience increasingly less deceleration at all times than the lower ball; except at launch where they briefly both feel the same deceleration. Under these conditions the balls would accelerate apart slightly throughtout their entire journey. But if g were to remain constant their separation would be constant.
 Quote by Idjot What I'd like to know now is why they would both decelerate at the same rate when one has more momentum.
Momentum only becomes a consideration where the force is not gravity. Even if the force is not gravity then momentum only becomes a consideration when the objects have different masses. If they are the same mass then the same force will change their speeds by the same amount.
We are dealing with gravity so momentum has no differential effect between the two balls in our examples. They could be two different masses and it wouldn't matter.

Gravity force acts directly upon the speed of objects the same amount no matter what their mass is. Only the amount of gravity is important. And it acts on the speeds - even if the speeds are different - by the same amount. So deceleration of the speed under a constant gravity force will be by a constant amount.

If you have a frictionless car travelling at 100 km/hr and it takes 1 unit of force to slow it down to 90 km/hr then it will also only take another 1 unit of force to slow it down further to 80 km/hr, and so on. The momentum at different speeds for the same mass makes no difference to this. The same amount of force (over any amount of time) changes the speed by the same amount.

Remember that force acts directly upon the speed something is going at - the change in speed being dependent upon its mass; except under gravity. Force does not act directly on the distance travelled. ie.

speed = distance / time ie. (L/T)
acceleration = force / mass ie. ((ML/T^2) / M = L/T^2)
speed2 = speed1 + acceleration * time ie. ((L/T) + (L/T^2) * T) = (L/T) + (L/T) = (L/T)

where L is a unit of length, T is a unit of time and M is a unit of mass,
where force is ML/T^2 and acceleration is L/T^2.

As I said under gravity the mass makes no difference so the above middle formula for acceleration would be different. Have to go but I will think about that; unless someone wants to quickly enlighten me too.