Whats the most efficient way to get vertical linear motion from an electric motor?
The most commonly used arrangement is the crank and pison (slide) arrangement which gives reciprocating linear motion from rotary motion. As you may already be aware this is the arrangement used in IC engines (internal combustion) both in-line and V type. Examples of IC engines are automobile engines, small aircraft engines (not common), farming appliances etc. Other common applications include reciprocating air compressors, refrigeration compressors, Air conditioning plants and electrical gensets. Below is a very simple diagram of such a mechanism. HOWEVER As you ask for the most effecient way for converting the rotary to linear motion then it is the rack and pinion mechanism. But the problem is this is cannot be a prolonged continuous process because it has to stop when the rack is finished i.e. it is limited by the rack lenght. Please see the 2nd image for this.