Disabling electric shopping cart upon exiting the store.

The store I work at provides electric shopping carts for customers to use. The problem is, they tend to be left outside and get rain damaged which is a $100-$200 fix every time. Here is what I'm thinking: Place either an IR or ultrasonic transmitter near the exit door and the matching receiver on each cart. When the cart gets too close to the door, have an Arduino flip a switch to break the power going to the motor.
What would I need to implement this cheaply and effectively? It looks like the sensors rely on sending a signal out and waiting for a response like echolocation, and not just a one-way signal.

Short range communication can be implemented in various ways (IR, RFID, BT...). I also could imagine an induction loop, like used for vehicle detection. That loop could be used not only to stop the motor, but also could be used as a rail where the cart can return into its safe harbour. Such a system can be extended into the parking place, if one exists, so that emptied carts can return autonomously.

For further ideas you should describe the cart more detailed, its capabilities and intended use.

I've seen conventional carts with a special electronic wheel that will lock up if it's taken outside without first going past a checkout. I'm guessing there's a transmitter buried under the door and one in the checkout.

How do the users of the self-propelled carts get their items to their cars? Are there attendants to help carry items? Ask them to drive the carts back. How do the conventional carts get back inside?

I'd modify the cart to just make an alarm sound if it's left for a long period of time outside (either via light sensor or not-plugged-in-to-charger) so that your staff can go out and find the lost cart. Add a moisture sensor so it screams louder during rain.

It would be fun if the carts were able to drive themselves back to the charging/pickup station but there's so many liability issues with that that the technology hasn't been implemented yet. My 10-year-old Roomba does, so why not a shopping cart?