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The velocity of an object is the rate of change of its position with respect to a
frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

, and is a function of time. Velocity is equivalent to a specification of an object's
speed In everyday use and in kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

and direction of
motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its position (mathematics), position over time. Motion is mathematically described in terms of Displacem ...
(e.g. to the north). Velocity is a fundamental concept in
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes it ...

, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ...
; both magnitude and direction are needed to define it. The
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

(
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
) of velocity is called , being a coherent derived unit whose quantity is measured in the SI (
metric system The metric system is a that succeeded the decimalised system based on the introduced in France in the 1790s. The historical development of these systems culminated in the definition of the (SI), under the oversight of an international stan ...

) as
metres per second The metre ( Commonwealth spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure", and cognate with Sanskrit Sanskrit (, attributively , ''saṃskṛta-'', nominalization, no ...
(m/s or m⋅s−1). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an ''
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

''.

# Constant velocity vs acceleration

To have a ''constant velocity'', an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration.

# Difference between speed and velocity

Speed, the
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
magnitude of a velocity vector, denotes only how fast an object is moving. Earliest occurrence of the speed/velocity terminology.

# Equation of motion

## Average velocity

Velocity is defined as the rate of change of position with respect to time, which may also be referred to as the ''instantaneous velocity'' to emphasize the distinction from the average velocity. In some applications the average velocity of an object might be needed, that is to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, , over some time period . Average velocity can be calculated as: :$\boldsymbol = \frac .$ The average velocity is always less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time (''x'' vs. ''t'') graph, the instantaneous velocity (or, simply, velocity) can be thought of as the , and the average velocity as the slope of the
secant line In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant inters ...
between two points with ''t'' coordinates equal to the boundaries of the time period for the average velocity. The average velocity is the same as the velocity averaged over time – that is to say, its time-weighted average, which may be calculated as the time integral of the velocity: :$\boldsymbol = \int_^ \boldsymbol\left(t\right) \ dt ,$ where we may identify :$\Delta \boldsymbol = \int_^ \boldsymbol\left(t\right) \ dt$ and :$\Delta t = t_1 - t_0 .$

## Instantaneous velocity

If we consider as velocity and as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time , as the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

of the position with respect to time: :$\boldsymbol = \lim_ \frac = \frac .$ From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time ( vs. graph) is the displacement, . In calculus terms, the
integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the velocity function is the displacement function . In the figure, this corresponds to the yellow area under the curve labeled ( being an alternative notation for displacement). :$\boldsymbol = \int \boldsymbol \ dt .$ Since the derivative of the position with respect to time gives the change in position (in
metre The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English o ...
s) divided by the change in time (in
second The second (symbol: s, also abbreviated: sec) is the of in the (SI) (french: Système International d’unités), commonly understood and historically defined as of a – this factor derived from the division of the day first into 24 s, th ...
s), velocity is measured in
metres per second The metre ( Commonwealth spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure", and cognate with Sanskrit Sanskrit (, attributively , ''saṃskṛta-'', nominalization, no ...
(m/s). Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment.

## Relationship to acceleration

Although velocity is defined as the rate of change of position, it is often common to start with an expression for an object's
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

. As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

of the to the curve of a graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time: :$\boldsymbol = \frac .$ From there, we can obtain an expression for velocity as the area under an acceleration vs. time graph. As above, this is done using the concept of the integral: :$\boldsymbol = \int \boldsymbol \ dt .$

### Constant acceleration

In the special case of constant acceleration, velocity can be studied using the
suvat equations In physics, equations of motion are equations that describe the behavior of a physical system in terms of its Motion (physics), motion as a function (mathematics), function of time.''Encyclopaedia of Physics'' (second Edition), Rita G. Lerner, R.G ...
. By considering a as being equal to some arbitrary constant vector, it is trivial to show that :$\boldsymbol = \boldsymbol + \boldsymbolt$ with as the velocity at time and as the velocity at time . By combining this equation with the suvat equation , it is possible to relate the displacement and the average velocity by :$\boldsymbol = \frac t = \boldsymbolt.$ It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows: :$v^ = \boldsymbol\cdot\boldsymbol = \left(\boldsymbol+\boldsymbolt\right)\cdot\left(\boldsymbol+\boldsymbolt\right) = u^ + 2t\left(\boldsymbol\cdot\boldsymbol\right)+a^t^$ :$\left(2\boldsymbol\right)\cdot\boldsymbol = \left(2\boldsymbol\right)\cdot\left(\boldsymbolt + \tfrac \boldsymbolt^\right) = 2t \left(\boldsymbol \cdot \boldsymbol\right) + a^t^ = v^ - u^$ :$\therefore v^ = u^ + 2\left(\boldsymbol\cdot\boldsymbol\right)$ where etc. The above equations are valid for both Newtonian mechanics and
special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words, only relative velocity can be calculated.

## Quantities that are dependent on velocity

The
kinetic energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
of a moving object is dependent on its velocity and is given by the equation :$E_ = \tfracmv^$ ignoring
special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, where ''E''k is the kinetic energy and ''m'' is the mass. Kinetic energy is a scalar quantity as it depends on the square of the velocity, however a related quantity,
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or w ...

, is a vector and defined by :$\boldsymbol=m\boldsymbol$ In
special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, the dimensionless
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativi ...

appears frequently, and is given by :$\gamma = \frac$ where γ is the Lorentz factor and ''c'' is the speed of light.
Escape velocity#REDIRECT Escape velocity In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), ...
is the minimum speed a ballistic object needs to escape from a massive body such as Earth. It represents the kinetic energy that, when added to the object's gravitational potential energy, (which is always negative) is equal to zero. The general formula for the escape velocity of an object at a distance ''r'' from the center of a planet with mass ''M'' is :$v_ = \sqrt = \sqrt,$ where ''G'' is the and ''g'' is the
Gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing air drag, drag). This is the steady gain in speed caused exclusively by the force of ''gravitational attraction' ...
. The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it doesn't intersect with something in its path.

# Relative velocity

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with
special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
in which velocities depend on the choice of reference frame. If an object A is moving with velocity
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
''v'' and an object B with velocity vector ''w'', then the velocity of object A ''relative to'' object B is defined as the difference of the two velocity vectors: :$\boldsymbol_ = \boldsymbol - \boldsymbol$ Similarly, the relative velocity of object B moving with velocity ''w'', relative to object A moving with velocity ''v'' is: :$\boldsymbol_ = \boldsymbol - \boldsymbol$ Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest.

## Scalar velocities

In the one-dimensional case,Basic principle
/ref> the velocities are scalars and the equation is either: :$v_\text = v - \left(-w\right)$, if the two objects are moving in opposite directions, or: :$v_\text = v -\left(+w\right)$, if the two objects are moving in the same direction.

# Polar coordinates

In
polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, a two-dimensional velocity is described by a
radial velocity The radial velocity of an object with respect to a given point is the rate of change of the distance between the object and the point. That is, the radial velocity is the component of the object's velocity that points in the direction of the radius ...
, defined as the component of velocity away from or toward the origin (also known as ''velocity made good''), and an
angular velocity In physics, angular velocity (\boldsymbol or \boldsymbol), also known as angular frequency vector,(UP1) is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angu ...

, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The
transverse Transverse may refer to: *Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle *Transverse flute, a flute that is held horizontally *Euler force, Transverse force (or ''Euler force''), ...
velocity is the component of velocity along a circle centered at the origin. :$\boldsymbol=\boldsymbol_T+\boldsymbol_R$ where *$\boldsymbol_T$ is the transverse velocity *$\boldsymbol_R$ is the radial velocity. The ''magnitude of the radial velocity'' is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of the velocity vector and the unit vector in the direction of the displacement. :$v_R=\frac$ where $\boldsymbol$ is displacement. The ''magnitude of the transverse velocity'' is that of the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of the unit vector in the direction of the displacement and the velocity vector. It is also the product of the
angular speed Angular frequency ''ω'' (in radians per second), is larger than frequency ''ν'' (in cycles per second, also called Hertz, Hz), by a factor of . This figure uses the symbol ''ν'', rather than ''f'' to denote frequency. In \boldsymbol, such that :$\omega=\frac.$ \boldsymbol\, . The expression $mr^2$ is known as moment of inertia. If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.

# References

* Robert Resnick and Jearl Walker, ''Fundamentals of Physics'', Wiley; 7 Sub edition (June 16, 2004). .