Is it possible to capture motion trails (like throwing a ball or running) using a camera like the Nikon Coolpix S70?
If so, how do I do it?
Unfortunately, due to a lack of a manual or shutter priority mode on the S70, it may be difficult to capture motion in the way you describe.
If there are no 'scene' modes available that provide a long shutter speed your best bet may be to mount the camera on a tripod, disable the flash & set the ISO to its lowest (if possible) then shoot in a low light situation. This will hopefully force the camera into a long enough shutter speed to create some form of motion trail.
If you had a Canon point & shoot you could install CHDK on it and manually adjust the shutter speed. I don't think there is anything similar for Nikon cameras, unfortunately.