Good zoom lens for taking baby pictures?

I'm looking for a lens for my Sony Nex 5. I'll be buying the adapter piece that allows me use other 3rd party lenses. I'm looking for a good telephoto zoom lens, in the 55-200 or 70-300 range, not sure which is best. I mainly take pictures of my 10 month old son. I've looked at different brands from Nikon to Canon and Tamron to Sigma, but have no clue what to get! I want something that's pretty quick to focus as a 10 month old moves a lot and rather quickly, I don't want to miss a shot! I also want good zoom. I'm looking to spend under $200.

Added (1). Not sure if it autofocuses.
I should add that I'm really just starting in photography and am not looking for something that fancy. I just want to be able to take pictures with good zoom. We're going on vacation back home and I want to be able to take pictures from the beach of my husband and son playing in the ocean.
Also, I did some more googling and noticed that VR and IS, the vibration reduction options of Canon and Nikon are not that useful. Is this true? I don't want to spend more money for those options if I never use them. I thought they would help with the objects that move fast, but from what I understand it's only if you shake.

Under $200 there aren't many options. I think the 70-300 is the only one and it's not the greatest quality to say the least.
You may want to double check and make sure you will have auto focus and metering when using those adapters. Usually you don't have at least one of those if not both.

I shoot Nikon. We're about to have twins and I think my Nikon AF-S 55-200 VR will do fine. I bought mine refurbished for $180 from Adorama about a year and a half ago. They are cheaper now. New for $190(http://www.adorama.com/NK55200VRWB.html) and refurbished for $140(http://www.adorama.com/NK55200VRR.html). If your camera uses sensor based stabilization, and I think it does, then you don't need the VR (vibration reduction). You can get some of the 70-300mm lenses starting at about $100. Nikon makes one that is selling for $155 new at Adorama(http://www.adorama.com/NK70300AFGU.html). The 55-200 is a better lens.

I'm wondering how you plan to focus with the lens. Many adapters do not support auto focus or metering. If you are missing either of these things you may be better off buying a lens made for your camera that will support these features. If you are trying to focus manually or if you need to deal with manual camera settings you could easily miss the shot.

I know you want a zoom, but another bargain lens is the Nikon 50mm F1.8(I paid $70 used). Nikon, Canon, Minolta, and Pentax all made lenses that still work with today's cameras that you should be able to find for under $100. I will caution you again to check to see if you would have auto focusing and metering. You could miss a lot of shots while playing around with your camera.

To begin with, you are not going to be buying any kind of digital zoom lens for less than 200 bucks! So you can scratch that right off your list.next, add a minimum of 300 -400 bucks to add the vr is feature!

the most likely lens you might be able to get for less than 200 bucks, Maybe, if you, re lucky might be a 50mm f1.8 lens which will mean than instead of taking shots from the beachhouse, you will have to leave the house and walk across the beach to take pictures