Final answer:
The force that stops a rolling shopping cart on a smooth tile floor is kinetic friction, contrary to static friction which prevents initial movement.
Step-by-step explanation:
The force that would eventually bring a shopping cart to a stop as it rolls across a smooth tile floor, assuming it does not crash into anything first, is kinetic friction. While moving, the cart experiences kinetic friction, which acts opposite to the direction of motion and gradually slows down the cart until it comes to a stop. This is different from static friction, which prevents an object from starting to move. Newton's First Law of Motion is applicable here, as it states that an object will continue in a state of constant velocity unless acted upon by a net external force, which in this case, is kinetic friction.