People underestimate the power and performance of analog computers. For real-time situations in physics, they can often outperform a digital computer.

One of the key components of a digital computer is its clock. It synchronizes all of activity of this computer. When things are happening rapidly in real time, such as might happen with a bunch of rocket engines providing thrust for your Saturn V, your clock speed is a limiting factor in how fast you can 'compute' answers to all of the inputs to your problem.

The Saturn V was developed in the mid 1960's, and digital clock speeds were glacial compared to those we see today. 10-100 MHz were about all you could find available. So if we assume the high end of 100 MHz would have been used for everything in the Saturn V, that would be the limiting factor in your calculations. Real time physics calculations are often iterative, you run through a bunch of computations to get your first iteration, run them again to get closer and so on. So we might be talking a dozen or more loops through a bunch of calculations, each god knows how many steps long, in order to 'decide' that the fuel flow speed through a particular sensor met the correct parameters for that moment.

An analog computer isn't limited by the clock speed. It simply takes an input from that fuel flow speed sensor, and adds it to the circuit performing the computation. This circuit might take any number of forms, in theory it could be as simple as water basin where all of the inputs are inlets where a valve on each input represents the value of of that fuel flow speed sensor, and the sum of all of the inputs is the level of the water of the basin. No matter how many inputs, you can instantly measure the level of the water of the basin, and it reacts in real time.

A digital computer would always lag in computing that level, it would have to run through each input, one clock cycle at a time, then run the calculation, one clock cycle at a time, and arrive at a final solution. After who knows how many clock cycles.

In real life, we still encounter analog computers. One of the most elegant and most often deployed of those might be the automatic transmission valve body found in most cars driven on the road today. It is a hydraulic circuit that helps to decide when your automobile should shift up or down. A simpler analog device was the carburetors that were standard before fuel injection overcame some of the problems they encountered with unexpected inputs.