Elevator

Why is it dangerous to suddenly stop an elevator which is moving downward at a moderate speed?

Not sure what you mean by "moderate speed"? Elevators in the US have rated speeds between 50 fpm and 2000 fpm. At speeds of 150 fpm or less for traction type elevators stopping is near instantaneous in an emrergency, very simialr for direct acting hydraulic elevators.

Elevators are generally designed to minimize discomfort to their riders then at what point do people not get injured? Falling, buckled knees, wrenched backs etc. across all age groups.

An experiement - at what point, speed as a function of wall height, is there is discomfort or injury when you step off the wall to land on carpeting lets say?

The elevator industry has indicated 150 fpm.