It depends on what your goal is with that $2000.
Do you want to achieve:
Leads / Sales / $$$$
Branding & awareness
It's true that you will hear AdWords has nothing to do with the organics, however I have witnessed websites with no links or prior rankings eventually rank high in the serps. After they have been receiving quality traffic and get good quality scores once a campaign has been run a while, that traffic must send the right signals via analytics. However if you are going to be spending money on AdWords you are more than likely going to be optimizing for a good quality score, this in turn would probably optimize the site for organics anyway.
So I would say they are both related, I would always start with a PPC campaign to see if the keywords and the landing page are profitable. After all if it's not profitable or doesn't generate the traffic you expected then why waste time?