Best Option...
Refurb/Used Nikon D7000 = $750
Refurb/Used Nikon 70-300mm VR f/4.5-5.6G ED = $350
Total: $1100 ($100 over your limit.. but worth it)
You could go cheaper on the body (D90, D5100... etc) but its worth the extra $$$ for a better camera. The 70-300mm VR is a great lens for outdoor (daylight) sports. It has a super fast Auto Focus and is built extremely well. The 39 point AF system on the D7000 is good for sports. The 6fps will get the job done (but the buffer fills up quickly). The strong point of the D7000 is its ISO handling that will help shooting sports on cloudy days with non-f/2.8 lenses.
I would not get the Sony DSLR (Flame suit activating).... Sony makes great cameras with some good technology, but as a company they aren't doing that great. Remember that you are buying into a SYSTEM not just a simple camera. Nikon and Canon have a better back-catalog of products to build upon. Sony is putting all of its efforts into the NEX mirrorless cameras. I wouldn't be surprised to see the end of the Alpha DSLR soon.
The Nikon V1 is a great point-n-shoot camera with detachable lenses.. but that's about it. I've helped a few people use it for their kids soccer/baseball games and i wasn't impressed. IMHO, a cheap D3100 w/Kit lens would do a better job. It also has a smaller sensor that limits the DOF you can achieve.
I'd look into buying a system you can keep upgrading... rather then spending $1000 on a dead-end.
(*Edit*) Adding a picture taken with the D7000 and Nikon 70-300mm VR... Not a great picture but shows what that setup can do. If i remember correctly it was an overcast day and it had just stopped raining.. This shot is from the max @ 300mm.
1/1600 sec | ISO 500 | f/7.1 | 300mm