distance measurement open circuit
Consider using a string pot.
http://www.celesco.com/faq/cet.htm

Otherwise, to stick with your original idea, you could possibly use a transponder based ultrasound system. You would have an ultrasound transceiver on the box, and a transponder on the ground. Periodically, the transceiver would send out an ultrasound burst. The transponder on the ground would detect and reply. The round trip time is proportional to the distance. If you need extreme accuracy, you would want to somehow adjust for temperature. The speed of sound is affected by temperature (and pretty much only by temperature).

Good luck!