In sports photography, it's not so much the camera as it is the lens. A fast telephoto is essential, and is going to be more important than the camera. The number one sports telephoto lens is a 70-200mm f/2.8.
If you use the 55-300mm, you may find that it does not have enough light gathering capability for either low-light shots (indoor situations) or, even outdoors the shutter speed may be too low to get blur-free photos of fast moving objects.
Unfortunately, f/2.8 zoom lenses are not inexpensive.
The choices are the AF-S Nikon 70-200mm f/2.8 ($2,400), the Sigma 70-200mm f./2.8 ($1,400), or the NIkon AF 80-200mm f/2.8 ($1,200).
Both the more expensive Nikon and Sigma versions have a built-in focus motor, so they will autofocus on entry level DSLRs such as the D5100. But the Nikon 80-200mm will not autofocus as it requires the camera to have an in-body focus motor - which the D5100 lacks.
Here is when buying a better DSLR is cheaper in the long run. If you by a D7000, it has an in-camera focus motor, and will autofocus the Nikon 80-200mm f/2.8. Although the cost of the D7000 is $400 more than the D5100, you will still save $800 by buying the 80-200mm f/2.8 rather than the 70-200mm f/2.8.
While the 70-200mm f/2.8 does have a couple of advantages over the 80-200mm f/2.8; slightly faster autofocusing, and vibration reduction, few amateur photographers can justify the $1,200 additional cost of those two features.
The optical quality of the 70-200mm and 80-200mm are very close, and either one will produce very good images. The Sigma's optical quality, while still very good, is not quite as good as either Nikon lens.