Range rate defines a signed scalar value describing the time rate of change of the range (distance) between two locations.
Derivation
Given a differentiable vector
Let
with
The magnitude of the position vector
The quantity range rate is the time derivative of the magnitude (norm) of
Substituting (2) into (3)
Evaluating the derivative of the right-hand-side
using (1) the expression becomes
Since
With
The range rate is simply defined as
the projection of the observer to target velocity vector onto the