Need help deciding on next lens

I just bought my first DLSR { D7100 with a Nikon 18-105mm lens}, had an Olympus OM-1 film camera . I have 2 Daughters that play soccer ,volleyball ,and Ski. age 11 and 9 and want to take pics of them both portrait and spots related! I bought a Nikon AF VR 55-200mm f/4.-5.6 zoom for starters !

Sometimes with a limited budget it seems like a chess game!!!!!

You will want both of course but I would complete the kit lens pair first with a 55-200 AF-S VR....an excellent lens.
The 70-300 AF-S VR is a bit sharper and longer but a lot more money. I have and use both and the 55-200 is a pleasant surprise.....

With a limited budget the 50mm would be redundant in coverage since the 18-105 covers that range already. If you need speed then yes. But for sports you normally need distance. The 55-200 is an ok lens but is not the sharpest on the D7000. I had one when I first got my D7000 and was not to pleased. Depends on your budget what you could get instead. The one I really love is the 70-300. Much sharper and just a little longer.

I am sure there are others that can give you some alternatives.

70-300 in either Tamron or Nikon versions. Tamron is great due to it's refined VC system on that lens. However this only if you are going to be shooting outdoor events in moderate natural lighting. 50 1.8 would be good for shooting outdoors or indoors with less than favorable lighting conditions. However if the budget allows try and look into something like a 70-200 F2.8 for low lighting conditions if you need do not need as much reach or the F4 version for outdoor shooting.

Thanks for the helpful info and ideas !! I found several in the Nikon line . The 70-300mm is available in 2 examples one with VR and one not for much less . With longer zooms I assume the VR is what I want! I did find a Nikon zoom in between price wise a
55-300mm VR DX and wanted to get opinions about selecting it over the 70-300 VR DX . It was $130-140 less.

The 55-300 is the DX version. While the 70-300 (FOV equivalent of 105-450 on DX cameras) are versions which are more/less suited towards FX cameras.