What are the different methods to covert rotational to linear motion in the example below?
The servo motor will rotate causing the metal rod to move in a linear motion as seen below.
Linear motion change is will be in the order of milliseconds
I believe the device you're looking for is called a leadscrew:
A leadscrew (or lead screw), also known as a power screw or translation screw, is a screw used as a linkage in a machine, to translate turning motion into linear motion.
The article goes on to list ball screws and roller screws as alternatives to the leadscrew, but I'm in the group of people that would call those types of leadscrews.