Suggestion: Crank up your ISO as far as it will go, and then back off one notch. Now see what kind of exposure solution your camera will do in sports mode. The shutter speed should be significantly faster than what you were trying.
Also think about depth of field. If you are shooting your daughter on a rope swing, then either you are on the side or else in front. If you are in front, then she is swinging in and out of the optimal focus, so you have to widen the depth of field significantly (which means smaller aperture, which means slower shutter, which is not what you want). If you are shooting from the side, then she is swinging along a constant plane of focus (which means you can get by with a small aperture, which means a faster shutter, which is what you want).
On the other hand, if you are in front and she is swinging toward you, then her apparent motion blur will be behind her and not so visible, but then you are back into the quandry above.
---Bob Gross---