Well, she was running from the police. That would make on suspicious of her intent. Maybe she kidnapped the child.
After she decided to leave, she ran several stop signs, and apparently went speeding through the school zone. In that case, you have to consider whether or not her driving constitutes a positive hazard, and she needed to be stopped, as opposed to merely tracked down later. It's a tough call, and could go either way -- but given that the cop didn't really know her intentions, a cop might reasonably assume that this lady, who's obviously disturbed, might go so far as to try to kill herself and her kid by forcing a head-on collision with some other driver. (I guess it's a somewhat popular form of suicide.)
This is a seriously disturbed lady, and I think Dear Old Dad probably bears some looking-into as well.
I really feel for the poor little kid. At school he's got to live with the stigma of having a crazy mom, and at home he's got to live with a crazy mom (and maybe dad).