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The missile knows where it is at all times
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So how are they supposed to know which missile will hit them and which will not wait and find out?
While the missile is gaining altitude, they can use radar data to determine when it is going fast enough on the current trajectory to likely reach them, then at a later point determine if it is high enough and fast enough that it will likely go past them. The time between those two points is when they have to sweat while telling the general population to seek shelter.
The missile knows where it is at all times
It knows this because it knows where it isn't. By subtracting where it is from where it isn't, or where it isn't from where it is, whichever is greater, it obtains a difference, or deviation.