Relative Velocity

Relative velocity (also or ) is the velocity of an object or observer B in the rest frame of another object or observer A.{\displaystyle {\vec {v}}_{b\middle a}}{\ displaystyle {\ vec {v}} _ {BA}}{\ displaystyle {\ vec {v}} _ {B \ operatorname {rel} A}}

Classical mechanics

In one dimension (non-relativistic)

Relative Velocity
Relative motion man in train

We start with relativistic motion in the classical , (or non- relativistic , or Newtonian approximation ) that all motion is much less than the speed of light. This limit is associated with the Galilean transformation . The figure shows a man on top of a train, on the back edge. At 1:00 pm he starts walking at the speed of 10 km/h (kmph). The train is running at a speed of 40 km/h. The figure shows the man and the train at two different times: first, when the journey began, and an hour later at 2:00 pm. The figure shows that the man is 50 km away from the starting point after traveling for one hour (by walking and by train). This is, by definition, 50 km/h, which suggests that the two velocities have to be added to calculate the relative velocity in this way.

The figure displays clocks and rulers to remind the reader that although the logic behind this calculation seems innocent, it creates a false impression of how clocks and rulers behave. ( See the train-and-platform thought experiment .) To recognize that this classical model of relativistic motion violates special relativity , we generalize the example to an equation:

{\displaystyle \underbrace {{\vec {v}}_{M\mid E}} _{\text{50 km/h}}=\underbrace {{\vec {v}}_{M\mid T}} _{\text{10 km/h}}+\underbrace {{\vec {v}}_{T\mid E}} _{\text{40 km/h}},}

Where from:

{\displaystyle {\vec {v}} _ {M \mid E}}~~Velocity ~~of ~~M ~~is ~~relative~~ to~~ E ~meaning,\\
{\displaystyle {\vec {v}} _ {M \mid T}}~~Velocity T ~~of ~~M~~ is~~ relative ~~to~~ rain,\\
{\displaystyle {\vec {v}} _ {T \mid E}}T~~ is~~ the ~~velocity ~~of ~~rain~~ with~~ respect ~~to ~~E ~mean .

Perfectly valid expressions for “velocity of A relative to B” include “velocity of A with respect to B” and “velocity of A in a coordinate system where B is always at rest”. Special relativity is violated because this equation for relativistic velocity falsely predicts that given the speed of light, different observers will measure different speeds.

In two dimensions (non-relative)

Relative Velocity
Relative velocity between two particles in classical mechanics

The figure shows two objects A and B moving with a constant velocity. The equations of motion are:

{\displaystyle {\vec {r}} _ {A} = {\vec {r}} _ {Ai} + {\vec {v}} _ {A} t,}
{\displaystyle {\vec {r}} _ {B} = {\vec {r}} _ {Bi} + {\vec {v}} _ {B} t,}

where the subscript i refers to the initial displacement (equal to zero at time t ). The difference between the two displacement vectors, , represents the location of B as seen from A.

{\vec r} _ {B} - {\vec r} _ {A}
{\displaystyle {\vec {r}}_{B}-{\vec {r}}_{A}=\underbrace {{\vec {r}}_{Bi}-{\vec {r}}_{Ai}} _{\text{initial separation}}+\underbrace {({\vec {v}}_{B}-{\vec {v}}_{A})t} _{\text{relative velocity}}.}

That’s why:

{\displaystyle {\vec {v}} _ {B \mid A} = {\vec {v}} _ {B} - {\vec {v}} _ {A}.}

After substituting and , you have:

{\vec v} _ {{A | C}} = {\vec v} _ {A}{\vec v} _ {{B | C}} = {\vec v} _ {B}
{\displaystyle {\vec {v}}_{B\mid A}={\vec {v}}_{B\mid C}-{\vec {v}}_{A\mid C}\Rightarrow }   {\displaystyle {\vec {v}}_{B\mid C}={\vec {v}}_{B\mid A}+{\vec {v}}_{A\mid C}.}

Galilean transformation (non-relativistic)

To make the theory of relativistic motion consistent with the theory of special relativity, we have to adopt a different convention. Continuing to work in the (non-relativistic) Newtonian limit , we begin with a Galilean transformation in one dimension:


where x’ is the position observed by a reference frame that is moving at speed, v, in the “unprimed” (x) reference frame. [Note 3] Taking the difference between the first of the two equations above, we have , , and what may seem like the obvious statement that , has:

{\displaystyle dx'=dx-v\,dt}dt'=dt
{\frac {dx'}{dt'}}={\frac {dx}{dt}}-v

To recover previous expressions for relative velocity, we assume that particle A is following a path defined by dx/dt in the unprimed context (and hence dx / dt in the primed frame). Thus and , where is and refer to the motion of A as observed by an observer in the unprimed and primed frame respectively. Recall that v is the motion of a stationary object in the primed frame, as seen from the unprimed frame. Thus we have , and:

{\displaystyle dx/dt=v_{A\mid O}}{\displaystyle dx'/dt=v_{A\mid O'}}OO'
{\displaystyle v = v_ {O '\mid O}}
{\displaystyle v_{A\mid O'}=v_{A\mid O}-v_{O'\mid O}\Rightarrow v_{A\mid O}=v_{A\mid O'}+v_{O'\mid O},}

where is the desired (easily learned) symmetry of the latter form.

Special relativity

As in classical mechanics, relative velocity in special relativity is the velocity of an object or observer B in the rest frame of another object or observer A. However, unlike in the case of classical mechanics, in special relativity, this is not usually the case .

{\vec {v}} _ {{\mathrm {B | A}}}
{\vec {v}} _ {{\mathrm {B | A}}} = - {\vec {v}} _ {{\mathrm {A | B}}}

This strange lack of symmetry is related to Thomas precedence and the fact that two consecutive Lorentz transformations rotate the coordinate system. This rotation has no effect on the magnitude of a vector, and therefore the relative motion is symmetric.

\ | {\vec {v}} _ {{\mathrm {B | A}}} \ | = \ | {\vec {v}} _ {{\mathrm {A | B}}} \ | = v_ { {\mathrm {B | A}}} = v _ {{\mathrm {B | A}}}

Parallel velocity

In the case where two objects are traveling in parallel directions, the relativistic formula for relative velocity is the same as the formula for adding relative velocities.

{\vec  {v}}_{{\mathrm  {B|A}}}={\frac  {{\vec  {v}}_{{\mathrm  {B}}}-{\vec  {v}}_{{\mathrm  {A}}}}{1-{\frac  {{\vec  {v}}_{{\mathrm  {A}}}{\vec  {v}}_{{\mathrm  {B}}}}{c^{2}}}}}

The relative speed is given by the formula:

v_{{\mathrm  {B|A}}}={\frac  {\left|v_{{\mathrm  {B}}}-v_{{\mathrm  {A}}}\right|}{1-{\frac  {v_{{\mathrm  {A}}}v_{{\mathrm  {B}}}}{c^{2}}}}}

vertical velocity

In the case where two objects are traveling in perpendicular directions, the relative relative velocity is given by the formula:

{\vec {v}} _ {{\mathrm {B | A}}}
{\vec  {v}}_{{\mathrm  {B|A}}}={{\frac  {{\vec  {v}}_{{\mathrm  {B}}}}{\gamma _{{\mathrm  {A}}}}}}-{\vec  {v}}_{{\mathrm  {A}}}

Where from

{\displaystyle \gamma _{\mathrm {A} }={\frac {1}{\sqrt {1-\left({\frac {v_{\mathrm {A} }}{c}}\right)^{2}}}}}

Relative speed is given by the formula

{\displaystyle v_{\mathrm {B|A} }={\frac {\sqrt {c^{4}-\left(c^{2}-v_{\mathrm {A} }^{2}\right)\left(c^{2}-v_{\mathrm {B} }^{2}\right)}}{c}}}

general case

The general formula for the relative velocity of an object or observer B in the rest frame of another object or observer A is given by the formula:

{\vec {v}} _ {{\mathrm {B | A}}}
{\displaystyle {\vec {v}}_{\mathrm {B|A} }={\frac {1}{\gamma _{\mathrm {A} }\left(1-{\frac {{\vec {v}}_{\mathrm {A} }{\vec {v}}_{\mathrm {B} }}{c^{2}}}\right)}}\left[{\vec {v}}_{\mathrm {B} }-{\vec {v}}_{\mathrm {A} }+{\vec {v}}_{\mathrm {A} }(\gamma _{\mathrm {A} }-1)\left({\frac {{\vec {v}}_{\mathrm {A} }\cdot {\vec {v}}_{\mathrm {B} }}{v_{\mathrm {A} }^{2}}}-1\right)\right]}

Where from

{\displaystyle \gamma _{\mathrm {A} }={\frac {1}{\sqrt {1-\left({\frac {v_{\mathrm {A} }}{c}}\right)^{2}}}}}

Relative speed is given by the formula

{\displaystyle v_{\mathrm {B|A} }={\sqrt {1-{\frac {\left(c^{2}-v_{\mathrm {A} }^{2}\right)\left(c^{2}-v_{\mathrm {B} }^{2}\right)}{\left(c^{2}-{\vec {v}}_{\mathrm {A} }\cdot {\vec {v}}_{\mathrm {B} }\right)^{2}}}}}\cdot c}