Will the universe collapse on itself

The ball analogy is actually quite similar, in a rather non-trivial way, to the way the expansion of the universe was thought of until the last decade or so. So, let me try to build the analogy.

If gravity as we conventionally think of it (massive bodies attracting each other) is really the only effect involved in the expansion of the universe, then we should expect that the expansion will be constantly slowing down. This does not, however, mean that it will necessarily reverse. To see this, let's consider your thrown ball.

If you throw a ball straight up, it will start slowing down immediately upon leaving your hand. Eventually it will reach some height where it stops and starts falling. We understand this by talking about energy. When the ball left your hand it had some amount of kinetic energy due to you throwing it. But, the higher an object gets above the earth, the more potential energy it must have to be there. Since energy is conserved, the only way for it to get potential energy is to take it away from the kinetic. If we use the standard convention that potential energy is 0 when you are infinitely far away from a massive object (so that an object at rest where it is not under the influence of gravity has 0 energy, not counting its mass), the formula for gravitation potential energy looks like PE = -MG/r, where M is the mass of the object whose gravity we're discussing (so, here, Earth) and r is the distance from the center of mass of that object.

You should be able to see that the difference between the potential energy at any place other than r=0 differs by only a finite amount from the potential energy an infinite distance away. Now, imagine you were to throw the ball at exactly the speed such that its kinetic energy is that difference (this speed is called "escape velocity"). Then, the ball would have to get infinitely far away before it stopped and turned around. And, if you threw the ball even faster, it wouldn't stop, even if it got infinitely far away, even though it would keep slowing down.

So, there are three possibilities for what happens to the ball, based on how fast you throw it. Either it is moving slower than escape velocity, and it will eventually fall back down; it is moving at escape velocity and it will never come back, but will be exactly at rest when infinitely far away; or it is moving faster than escape velocity, and will never even come to rest.

The simple model of the expansion of the universe actually has these same three possibilities, based on the amount of mass in the universe and the initial rate of expansion. If the universe isn't expanding fast enough, based on the density of mass present, it should eventually collapse. If these factors are exactly balanced, the expansion will slow exactly to a stop after an infinite time. And, if the expansion rate is faster than that, it will never stop expanding.

However, observations in the mid to late '90s threw a serious wrench in the works for these models. They saw things that only made sense if the expansion was speeding up, not slowing down at all. This is not something that can be understood in terms of any simple example; and, in fact, the physics behind it is not particularly well understood yet.

Looking at the equation of general relativity, as applied to the problem of Cosmology, we can derive some properties of a material (ominously referred to as "dark energy"), the presence of which would cause the expansion to acceleration; but, it's not clear from the rest of our understanding of physics exactly what this should be.

We do know, however, the dark energy must have some very unusual properties. First, its energy density should not decrease as the universe expands. This means that if a patch of space doubles in volume, the amount of dark energy in it will double as well. Also, the dark energy should have negative pressure.

Right now, the only candidate for dark energy is what is known as vacuum energy - the energy associated with the lowest energy (or ground) states of the quantum mechanical fields comprising every type of particle in existance (electrons, photons, neutrinos, etc.). Vacuum energy does have the property that its energy density should be unaffected by expansion. However, the effect vacuum energy would create is larger than the observed effect of dark energy by a factor of ~10^120. So, clearly, we have a long way to go in understanding dark energy.

So, in summary, the thrown ball model is in some ways similar to the question of the expansion of a universe where normal gravitational effects are the only ones present; but, that does not seem to be the case in our universe. And, even in the simple model, the universe does not have to collapse in the end.

As a final note, the energy density and rate of expansion of the universe are so close to the critical values corresponding to the "escape velocity" case that, were the simple model correct, we actually wouldn't know which fate the universe awaited.