Intercontinental ballistic missiles (ICBM) are first and foremost ballistic.
This means that they work by tossing the nuclear bomb at the target in just the same way a basketball player shoots a hoop. At some point early in the action, the ball leaves the player's hand with all the speed needed to make a nice arc into the basket. Just so, the warhead leaves the Minuteman early in its flight after all three stages have been expended.
Basic physics tells us that the speed and direction of the ball as it leaves your hand completely predicts whether you will score points or, in the case of Strategic Air Command, hit the target.
A Minuteman ICBM has an on-board computer to tell the booster when enough thrust has been generated because the speed is perfect. But what tells the computer what speed the missile is going?
Therein lies a tale.
The Germans appear to be the first to use the basic principle of a PIGA in a weapon, the V-2 rocket bomb employed against England. Dr. Fritz Mueller solved the problem of measuring speed by use of a Mueller Mechanical Integrating Accelerometer (MMIA). The precession of a gyroscope turns a valve to shut off the V-2's liquid fuel. See diagram above.
Mueller's device used a weight handing off the end of a spinning gyro to measure speed. This pendulous weight caused the mechanism to precess or spin. The physics of the gyro tells us that the number of rotations of the device is proportional to speed. Thus, the device senses acceleration with the weight, but the readout is speed. Math definitions tell us that speed is the integral of acceleration just as acceleration is the derivative of speed.
So the device was also called a pendulous integrating gyroscopic accelerometer or PIGA.
After WWII, Dr. Charles Stark Draper at MIT took the MMIA and improved its precision and accuracy immensely. Many early ICBM designers felt that the only way to ensure the required accuracy would be a remote control radio system. Dr. Draper and others thought this kind of control system would be far too susceptible to enemy spoofing. Draper and MIT's ability to prove to the government decision-makers that PIGAs were the answer was one of several technological turning points in the creation and deployment of today's ICBM and, ultimately, our ability to land men on the moon.
Speed is one thing. But direction is also critical. How does the on-board computer know the speed is in the right direction? Once the speed is met, how does the computer tell the third stage of the missile to stop thrusting? Remember, this is solid fuel and solid fuel cannot be metered through a valve like liquid fuel. Some Minuteman ICBMs throw more than one warhead. How can that work in this scheme?
All of that is a tale for another post.