As many others have already said, the AF-S 70-300 mm f/4-5.6 VR is the easy pick of a lens to get on a budget for sports, with nearly instant focusing and a reasonable size with VR. Two other ones that I find are a good match for the D5100 body:
- AF-S 300 mm f/4: It's basically the entry level super telephoto, but the price isn't so super (it's the same price as a 70-200 f/4, and much less expensive than a 70-200 f/2.8). Takes a TC-14E for extra reach, and on newer bodies (like the D5200) can even use a TC-20E to get you out to 600 mm. It's an older lens, but you won't be disappointed with it.
- AF-S 55-200 mm f/4-5.6 DX VR: Cheap and cheaply made, but it's sharp, and I returned my 55-300 because it didn't focus fast enough on my D5100 body. It's also about the biggest lens that you can get away with in many professional sporting venues, and forms a great kit walk-around kit with the 18-55 your camera probably came with.
- AF-S 70-200 mm f/4 VR: Also allows teleconverters like the 300 f/4, but has less reach zooms and has VR, for about the same price, and it doesn't break the bank quite like the 70-200 f/2.8 VR.