This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Relative velocity" – news ·newspapers ·books ·scholar ·JSTOR(November 2024) (Learn how and when to remove this message)

Therelative velocity of an objectB with respect to an observerA, denoted${\displaystyle {\mathbf{v}}_{B\mid A}}$ (also${\displaystyle {\mathbf{v}}_{BA}}$ or${\displaystyle {\mathbf{v}}_{B\mathrm{rel}A}}$), is thevelocityvector ofB measured in therest frame ofA.Therelative speed is thevector norm of the relative velocity,${\displaystyle v_{B\mid A}=\Vert {\mathbf{v}}_{B\mid A}\Vert }$.

We begin with relative motion in theclassical, (or non-relativistic, or theNewtonian approximation) that all speeds are much less than the speed of light. This limit is associated with theGalilean transformation. The figure shows a man on top of a train, at the back edge. At 1:00 pm he begins to walk forward at a walking speed of 10 km/h (kilometers per hour). The train is moving at 40 km/h. The figure depicts the man and train at two different times: first, when the journey began, and also one hour later at 2:00 pm. The figure suggests that the man is 50 km from the starting point after having traveled (by walking and by train) for one hour. This, by definition, is 50 km/h, which suggests that the prescription for calculating relative velocity in this fashion is to add the two velocities.

The diagram displays clocks and rulers to remind the reader that while the logic behind this calculation seem flawless, it makes false assumptions about how clocks and rulers behave. (SeeThe train-and-platform thought experiment.) To recognize that thisclassical model of relative motion violatesspecial relativity, we generalize the example into an equation:

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

where:

${\displaystyle {\mathbf{v}}_{M\mid E}}$ is the velocity of theMan relative toEarth,

${\displaystyle {\mathbf{v}}_{M\mid T}}$ is the velocity of theMan relative to theTrain,

${\displaystyle {\mathbf{v}}_{T\mid E}}$ is the velocity of theTrain relative toEarth.

Fully legitimate expressions for "the velocity of A relative to B" include "the velocity of A with respect to B" and "the velocity of A in the coordinate system where B is always at rest". Theviolation of special relativity occurs because this equation for relative velocity falsely predicts that different observers will measure different speeds when observing the motion of light.^{[note 1]}

The figure shows two objectsA andB moving at constant velocity. The equations of motion are:

${\displaystyle {\mathbf{r}}_A={\mathbf{r}}_{Ai}+{\mathbf{v}}_{A}t,}$

${\displaystyle {\mathbf{r}}_B={\mathbf{r}}_{Bi}+{\mathbf{v}}_{B}t,}$

where the subscripti refers to the initial displacement (at timet equal to zero). The difference between the two displacement vectors,${\displaystyle {\mathbf{r}}_B-{\mathbf{r}}_A}$, represents the location of B as seen from A.

${\displaystyle {\mathbf{r}}_B-{\mathbf{r}}_A=\underset{\text{initial separation}}{\underbrace{{\mathbf{r}}_{Bi}-{\mathbf{r}}_{Ai}}}+\underset{\text{relative velocity}}{\underbrace{({\mathbf{v}}_B-{\mathbf{v}}_A)t}}.}$

Hence:

${\displaystyle {\mathbf{v}}_{B\mid A}={\mathbf{v}}_B-{\mathbf{v}}_A.}$

After making the substitutions${\displaystyle {\mathbf{v}}_{A|C}={\mathbf{v}}_A}$ and${\displaystyle {\mathbf{v}}_{B|C}={\mathbf{v}}_B}$, we have:

${\displaystyle {\mathbf{v}}_{B\mid A}={\mathbf{v}}_{B\mid C}-{\mathbf{v}}_{A\mid C}\Rightarrow }$  ${\displaystyle {\mathbf{v}}_{B\mid C}={\mathbf{v}}_{B\mid A}+{\mathbf{v}}_{A\mid C}.}$

To construct a theory of relative motion consistent with the theory of special relativity, we must adopt a different convention. Continuing to work in the (non-relativistic)Newtonian limit we begin with aGalilean transformation in one dimension:^{[note 2]}

${\displaystyle x^{\prime }=x-vt}$

${\displaystyle t^{\prime }=t}$

where x' is the position as seen by a reference frame that is moving at speed, v, in the "unprimed" (x) reference frame.^{[note 3]} Taking the differential of the first of the two equations above, we have,${\displaystyle d{x}^{\prime }=dx-vdt}$, and what may seem like the obvious^{[note 4]} statement that${\displaystyle d{t}^{\prime }=dt}$, we have:

${\displaystyle \frac{d{x}^{\prime }}{d{t}^{\prime }}=\frac{dx}{dt}-v}$

To recover the previous expressions for relative velocity, we assume that particleA is following the path defined by dx/dt in the unprimed reference (and hencedx′/dt′ in the primed frame). Thus${\displaystyle dx/dt=v_{A\mid O}}$ and${\displaystyle d{x}^{\prime }/dt=v_{A\mid O^{\prime }}}$, where${\displaystyle O}$ and${\displaystyle O^{\prime }}$ refer to motion ofA as seen by an observer in the unprimed and primed frame, respectively. Recall thatv is the motion of a stationary object in the primed frame, as seen from the unprimed frame. Thus we have${\displaystyle v=v_{O^{\prime }\mid O}}$, and:

${\displaystyle v_{A\mid O^{\prime }}=v_{A\mid O}-v_{O^{\prime }\mid O}\Rightarrow v_{A\mid O}=v_{A\mid O^{\prime }}+v_{O^{\prime }\mid O},}$

where the latter form has the desired (easily learned) symmetry.

As in classical mechanics, in special relativity the relative velocity${\displaystyle {\mathbf{v}}_{\mathrm{B}|\mathrm{A}}}$ is the velocity of an object or observerB in the rest frame of another object or observerA. However, unlike the case of classical mechanics, in Special Relativity, it is generallynot the case that

${\displaystyle {\mathbf{v}}_{\mathrm{B}|\mathrm{A}}=-{\mathbf{v}}_{\mathrm{A}|\mathrm{B}}}$

This peculiar lack of symmetry is related toThomas precession and the fact that two successiveLorentz transformations rotate the coordinate system. This rotation has no effect on the magnitude of a vector, and hence relativespeed is symmetrical.

${\displaystyle \Vert {\mathbf{v}}_{\mathrm{B}|\mathrm{A}}\Vert =\Vert {\mathbf{v}}_{\mathrm{A}|\mathrm{B}}\Vert =v_{\mathrm{B}|\mathrm{A}}=v_{\mathrm{A}|\mathrm{B}}}$

In the case where two objects are traveling in parallel directions, the relativistic formula for relative velocity is similar in form to the formula for addition of relativistic velocities.

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

Therelative speed is given by the formula:

${\displaystyle v_{\mathrm{B}|\mathrm{A}}=\frac{\left|{\mathbf{v}}_{\mathrm{B}}-{\mathbf{v}}_{\mathrm{A}}\right|}{1-\frac{{\mathbf{v}}_{\mathrm{A}}{\mathbf{v}}_{\mathrm{B}}}{c^2}}}$

In the case where two objects are traveling in perpendicular directions, the relativistic relative velocity${\displaystyle {\mathbf{v}}_{\mathrm{B}|\mathrm{A}}}$ is given by the formula:

${\displaystyle {\mathbf{v}}_{\mathrm{B}|\mathrm{A}}=\frac{{\mathbf{v}}_{\mathrm{B}}}{{\gamma }_{\mathrm{A}}}-{\mathbf{v}}_{\mathrm{A}}}$

where

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

The relative speed is given by the formula

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

The general formula for the relative velocity${\displaystyle {\mathbf{v}}_{\mathrm{B}|\mathrm{A}}}$ of an object or observerB in the rest frame of another object or observerA is given by the formula:^[1]

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

where

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

The relative speed is given by the formula

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

• Doppler effect
• Non-Euclidean geometry § Kinematic geometries
• Peculiar velocity
• Proper motion
• Range rate
• Radial velocity
• Rapidity
• Relativistic speed
• Space velocity (astronomy)

• Alonso & Finn, Fundamental University PhysicsISBN 0-201-56518-8
• Greenwood, Donald T, Principles of Dynamics.
• Goodman and Warner, Dynamics.
• Beer and Johnston, Statics and Dynamics.
• McGraw Hill Dictionary of Physics and Mathematics.
• Rindler, W., Essential Relativity.
• KHURMI R.S., Mechanics, Engineering Mechanics, Statics, Dynamics

• Relative Motion at HyperPhysics
• A Java applet illustrating Relative Velocity, by Andrew Duffy
• Relatív mozgás (1)...(3) Relative motion of two train (1)...(3). Videos on the portalFizKapu.(in Hungarian)
• Sebességek összegzése Relative tranquility of trout in creek. Video on the portalFizKapu.(in Hungarian)

• Physical quantities
• Classical mechanics
• Special relativity
• Velocity

• Articles with short description
• Short description is different from Wikidata
• Articles needing additional references from November 2024
• All articles needing additional references
• Articles with Hungarian-language sources (hu)