If the lens was extended when the camera fell, it is very possible that the lens barrel has been knocked off center.
When you turn on the power, the first thing it does is try to extend the lens barrel. If it is slightly bent, it could be sticking a bit.
Any type of jarring could unstick it.
Look at the area around the lens before turning-on the camera and see if it is off center.
Look again with it extended and look for the same thing.
It could also be something loose inside.
Just about any repair by Panasonic will run over $100.
Looking for some electronics experts here!
My FX30 fell on the floor a few months ago, there did not appear to be any damage however. I used the camera a few times after, I then stored the camera for a few weeks.
Recently the camera developed a weird problem which could be related to the crash that occurred months before. The camera won't turn itself on, after checking pretty much every possible alternative I was ready to concede it was ready for the garbage can. In a gesture of frustration, I threw the camera on the table and bingo, it turned itself on and started working normally again. A few days later, again it wouldn't power on, I then banged it against the table a few times and sure enough, that was all it needed to get back to life. I've had to hit it quite forcefully to get it to work recently, it seems to be getting worse.
Any ideas? Some electrical connection that's loose? Perhaps the crash protection for internals that's been crushed? Some kind of spring loaded protection that's been broken? I do hear a very slight noise when I shake the camera, there could be something loose inside but again this sound may be entirely normal as it is very minor.
I'm sure taking the camera to Panasonic will cost an arm, so I'll wait for advice first.