Gravitationally bound and Hubble time

[BearY]

Homework Statement

Estimate how long a galaxy in the Coma cluster would take to travel from one side of the cluster to the other. Assume that the galaxy moves with a constant speed equal to the cluster’s radial velocity dispersion. How does this compare with the Hubble time, t H ? What can you conclude about whether the galaxies in the Coma cluster are gravitationally bound?

Carroll, Bradley W.; Ostlie, Dale A.. Introduction to Modern Astrophysics, An: Pearson New International Edition (Page 1184)

Homework Equations

The Attempt at a Solution

The first 2 questions are simple, with values(from some very not reliable encyclopedia)
diameter is 6 mpc
radial dispersion velocity is 1000km/h
hubble time = 13.8 billion years
I reached a conclusion of the time needed is about 0.4 Hubble time.

But I have no clue how does it imply anything about gravitational bound.

 

[mfb]

How does the motion of a galaxy look like if the timescale is much smaller than the Hubble time?
How does it look like if it is much larger?
In which case do you get something that looks as expected for a gravitationally bound object?

 

[BearY]

How does the motion of a galaxy look like if the timescale is much smaller than the Hubble time?
How does it look like if it is much larger?
In which case do you get something that looks as expected for a gravitationally bound object?

If the radial dispersion velocity is too large then the system is probably no gravitational bounded, because then the kinetic energy of most of the galaxies in the cluster would be enough to escape?
If it is very slow too slow I can't think of anything else except that it can be bounded?
Yesterday I had an idea about if it has anything to do with the virial theorem. We can assume the cluster is bounded and estimate the mass enclosed using Faber–Jackson relation, then with mass enclosed we can estimate the average Ek and average Ep, and see if it violate the virial theorem by too much?
But to be honest I don't know how does it relate to Hubble time and still pretty clueless in general.

 

[mfb]

Now you are overthinking this. The timescale you found is all you need.

As an example, for Earth the timescale is 1 year. It has been done billions of orbits. It is clearly gravitationally bound.
If you take two galaxy clusters far away you get a timescale of hundreds of billions of years: They cannot orbit each other. There was not even enough time in the universe for that.