Mom of HS Athlete

These cameras are very close in price (within $50 according to sortable), but the D5100 has considerably better image quality according to DXOMark.  If it was me I'd get the D5100.

That said, for your particular requirements of 'distant subjects' and 'fast action' the lens you buy will probably make more of a difference than the body.  You should check out http://lenshero.com/lenses/nikon-d5100-telephoto-lens to see what is available for the D5100.

Both cameras are "capable" of fast action, distant images but what's very much more relevant and important is the lens that goes on these cameras.  You'll need a "fast" lenses such as Nikon 70-200mm f/2.8 VR to do what you desire. And they are usually very expensive, anywhere from $1,200 to $2,400.  Third party lens makers such as Sigma and Tamron make cheaper versions of this fast lens but they are still expensive.

You might still be able to capture decent images when lighting is good with consumer level lenses such as 55-300mm or 70-300mm lenses. You'll want to get one with VR (vibration reduction) designation.

As far as camera goes, D5100 is a step up from D3100. 

Price is so close, d5100 has many advantages as listed by @Philip-Davis.  Good camera, great price.

The D5100 is, hands-down, a better camera than the D3100.  It has better image quality, better battery life, faster shooting and has a flip-out screen, which is handy for shooting video.