Are Saturn's rings a safe place for adolescent space-hoppers?

The rings are a cloud of many objects that orbit Saturn very quickly. By very quickly I mean kilometers per second. This is why the ring's don't fall down: The centrifugal force pushing them outwards balances with the gravitational force pulling them inwards. If the Hoppers live inside the rings them they are also orbiting very quickly and they experience weightlessness.

If, in their youthful exuberance, they bounce too far, could they escape the rings altogether?

Yes. If you stand on the edge of a ring and jump as hard as you can, you will drift outwards for a while. Then you slow down and eventually take up an orbit above the ring.

If there is nothing to push off against then you are trapped up there and will starve. One solution if for your friends to throw small rocks at you to catch. You then throw the rocks away from the planet to propel yourself back down. Unfortunately the hoppers don't have hands. . . .

Of course this assumes the rings have hard edges. Certainly looks like it huh?

Keep in mind this photo is from very VERY far away. The planet itself has no hard edge so there is no reason to believe the rings do. It's far more likely they have blurred edges where the density of the rocks tapers off. So bouncing into empty space is not much of a problem. There is always something to bounce back off.

Would they enter a new orbit and maybe meet up with the rings again eventually?

If there is a hard gap between two rings you might be able to jump across the gap. But if you don't jump hard enough you will get trapped between like before. I don't think it matters if you jump towards or away from the planet.

Is it possible to calculate escape velocity for the rings of Saturn?

Yes! It's real real big and Franklin has already done so.

Does it depend hugely on the direction of travel?

I don't think it makes much of a difference. The direction of travel is important if you're trying to achieve orbit since you need to be going around very quickly to generate the centrifugal force. Keep in mind most spacecraft don't actually escape the gravity well.

The escape velocity $v_e$ required to escape a body of mass $M$ from a distance $r$ is equal to

$$v_e=sqrt{frac{2GM}{r}}$$

And it happens that the escape velocity is the same in all directions. Regardless of whether your trajectory is directly away from the planet or oblique, a velocity of $v_e$ will send you off into space.

The mass of Saturn is about $Mapprox 5.68cdot 10^{26}spacetext{kg}$, and the gravitational constant $G$ is about $Gapprox 6.67cdot 10^{-11}spacetext{Nm}^2/text{kg}^2$. The radius of Saturn’s Phoebe ring (which is the farthest out of all its rings, and therefore the easiest to escape from) is about $rapprox 1.75cdot 10^{10}spacetext{m}$. This means that escape velocity from the ring system is about

$$v_e approx sqrt{frac{(2)(6.67cdot 10^{-11})(5.68cdot 10^{26})}{1.75cdot 10^{10}}}approx 2080.9spacetext{m/s}$$

The fastest land animal on Earth - the cheetah - can travel at a speed of $33.53 spacetext{m/s}$, which is much slower than the required velocity. And it doesn’t sound like your space-hoppers are anywhere near as fit as a cheetah.

However, there is another potential problem. Since Saturn’s rings are also moving at a constant velocity, the space-hoppers could possibly piggyback on the rings’ velocity to reach escape velocity.

The orbital speed of an object orbiting at a distance $r$ is approximately

$$v_o = sqrt{frac{GM}{r}} = frac{v_e}{sqrt{2}}$$

So we have that Saturn’s rings are moving at an orbital velocity of about

$$v_oapprox 1471.4spacetext{m/s}$$

This means that the extra velocity that space-hoppers must generate in order to escape is given by

$$v_e - v_o approx 609.5spacetext{m/s}$$

...which is still much faster (more than $18$ times faster) than a cheetah can travel. So it’s pretty unlikely that your space-hoppers can escape, unless they have extremely powerful spring-like appendages.

As for crashing into the planet - that’s always a danger, if the space-hoppers wander too close to the inner edge of the rings. If they propel themselves beyond this inner edge, there will be nothing for them to grab onto (not even an atmosphere to fly in) to halt their acceleration towards the planet. The best bet would be to stay near the outer edge of the rings, since the change of accidentally escaping is practically nil.

EDIT: @Daron has considered the possibility of escaping the ring system but not the planet’s orbit, taking up an orbit outside of the planet’s rings. The calculations above haven’t accounted for that possibility, just the possibility of flying out into space forever. Presumably, if a baby space-hopper flew into a more distant orbit, all hope would not be lost and its parents might devise away of drawing it back. Let’s deal with this situation numerically.

We can calculate the radius of the farthest possible orbit that a space-hopper can fling itself into using a calculation involving the energy of the orbit. The total energy of an orbiting object with mass $m$ at a distance $r$ from a planet of mass $M$ is given by

$$E=-frac{GMm}{2r}$$

The energy $dE$ required to change the orbit radius by a small amount $dr$ is about

$$dEapprox frac{GMm}{2r^2}cdot dr$$

Suppose that your space-hopper is about the mass of a cow, making it about $1.4cdot 10^6$ grams. Then the approximate energy required to increase the orbit radius by $1$ meter starting from the Phoebe ring is about $0.087spacetext{N}$. That’s not very much energy.

The bite force of an alligator is around $13,000spacetext{N}$, so if your space-hopper can jump as hard as an alligator can chomp, then it could increase the radius of its orbit by around $1.5cdot 10^{5}spacetext{m}$. Oof.

On second thought, better stay towards the middle of the rings.