Range rate - Alchetron, The Free Social Encyclopedia

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 r ϵ R 3 defining the instantaneous position of a target relative to an observer.

Let

with v ϵ R 3 , the instantaneous velocity of the target with respect to the observer.

The magnitude of the position vector r is defined as

The quantity range rate is the time derivative of the magnitude (norm) of r , expressed as

Substituting (2) into (3)

d | | r | | d t = d ⟨ r , r ⟩ 1 / 2 d t

Evaluating the derivative of the right-hand-side

d | | r | | d t = 1 2 d ⟨ r , r ⟩ d t 1 ⟨ r , r ⟩ 1 / 2 d | | r | | d t = 1 2 ⟨ d r d t , r ⟩ + ⟨ r , d r d t ⟩ ⟨ r , r ⟩ 1 / 2

using (1) the expression becomes

d | | r | | d t = 1 2 ⟨ v , r ⟩ + ⟨ r , v ⟩ ⟨ r , r ⟩ 1 / 2

Since

⟨ v , r ⟩ = ⟨ r , v ⟩

With

r ^ = r ⟨ r , r ⟩ 1 / 2

The range rate is simply defined as

d | | r | | d t = ⟨ r , v ⟩ ⟨ r , r ⟩ 1 / 2 = ⟨ r ^ , v ⟩

the projection of the observer to target velocity vector onto the r ^ unit vector.

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