The relative velocity
(also
or
) is the velocity of an object or observer B in the rest frame of another object or observer A.
Classical mechanics
In one dimension (non-relativistic)
![Relative motion man on train](https://upload.wikimedia.org/wikipedia/commons/3/37/Relative_motion_man_on_train.gif)
We begin with relative motion in the
classical, (or non-
relativistic, or the
Newtonian approximation
In physics, the Newtonian limit is a mathematical approximation applicable to physical systems exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly changing (or completely static) gravitational ...
) that all speeds are 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, 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. (See
The train-and-platform thought experiment.) To recognize that this
classical model of relative motion violates
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
, we generalize the example into an equation:
:
where:
:
is the velocity of the Man relative to Earth,
:
is the velocity of the Man relative to the Train,
:
is the velocity of the Train relative to Earth.
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". The
violation 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.
[For example, replace the "Man" by a photon traveling at the speed of light.]
In two dimensions (non-relativistic)
![Relative velocity](https://upload.wikimedia.org/wikipedia/commons/b/b5/Relative_velocity.svg)
The figure shows two objects ''A'' and ''B'' moving at constant velocity. The equations of motion are:
:
:
where the subscript ''i'' refers to the initial displacement (at time ''t'' equal to zero). The difference between the two displacement vectors,
, represents the location of B as seen from A.
:
Hence:
:
After making the substitutions
and
, we have:
:
Galilean transformation (non-relativistic)
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 a
Galilean transformation in one dimension:
[This result is valid if all motion is restricted to the x-axis, but can be easily generalized by replacing the first equation by ]
:
:
where x' is the position as seen by a reference frame that is moving at speed, v, in the "unprimed" (x) reference frame.
[It is easy to be confused about the minus sign before ''v'', or whether ''v'' is defined in the prime or unprimed reference frame. It might help to visualize the fact that if ''x'' = ''vt'', then ''x''′ = 0, meaning that a particle that is following the path ''x'' = ''vt'' is at rest in the primed reference frame.] Taking the differential of the first of the two equations above, we have,
, and what may seem like the obvious
[Keep in mind that, due to ]time dilation
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
, ''dt'' = ''dt''′ is valid only in the approximation that the speed is much less than that of light. statement that
, we have:
:
To recover the previous expressions for relative velocity, we assume that particle ''A'' is following the path defined by dx/dt in the unprimed reference (and hence ''dx''′/''dt''′ in the primed frame). Thus
and
, where
and
refer to motion of ''A'' as seen 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:
:
where the latter form has the desired (easily learned) symmetry.
Special relativity
As in classical mechanics, in Special Relativity the relative velocity
is the velocity of an object or observer B in the rest frame of another object or observer A. However, unlike the case of classical mechanics, in Special Relativity, it is generally not the case that
:
This peculiar lack of symmetry is related to
Thomas precession
In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a pa ...
and the fact that two successive
Lorentz transformations
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
rotate the coordinate system. This rotation has no effect on the magnitude of a vector, and hence relative
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (ma ...
is symmetrical.
:
Parallel velocities
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.
:
The relative speed is given by the formula:
:
Perpendicular velocities
In the case where two objects are traveling in perpendicular directions, the relativistic relative velocity
is given by the formula:
:
where
:
The relative speed is given by the formula
:
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:
[Fock 1964 The theory of Space Time and Gravitation, retrieved from https://archive.org/details/TheTheoryOfSpaceTimeGravitation]
:
where
:
The relative speed is given by the formula
:
See also
*
Doppler effect
*
*
Peculiar velocity
Peculiar motion or peculiar velocity refers to the velocity of an object relative to a ''rest frame'' — usually a frame in which the average velocity of some objects is zero.
Galactic astronomy
In galactic astronomy, peculiar motion refers to ...
*
Proper motion
*
Range rate
The radial velocity or line-of-sight velocity, also known as radial speed or range rate, of a target with respect to an observer is the rate of change of the distance or range between the two points. It is equivalent to the vector projection o ...
*
Radial velocity
*
Rapidity
*
Relativistic speed
*
Space velocity (astronomy)
In astronomy, stellar kinematics is the observational study or measurement of the kinematics or motions of stars through space.
Stellar kinematics encompasses the measurement of stellar velocities in the Milky Way and its satellites as well as ...
Notes
References
Further reading
* Alonso & Finn, Fundamental University Physics
* 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
External links
Relative Motion at HyperPhysics*
ttp://www.fizkapu.hu/fizfilm/fizfilm1.html Relatív mozgás (1)...(3)Relative motion of two train (1)...(3). Videos on the porta
FizKapu Relative tranquility of trout in creek. Video on the porta
FizKapu
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Physical quantities
Classical mechanics
Special relativity
Velocity