Velocity is the
directional 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 ...
of an
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
in
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
as an indication of its
rate of change in
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
as observed from a particular
frame of reference and as measured by a particular standard of time (e.g.
north
North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating direction or geography.
Etymology
The word ''north ...
bound). Velocity is a fundamental concept in
kinematics, the branch of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
that describes the motion of bodies.
Velocity is a physical
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
quantity
Quantity or amount 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 multiple of a unit ...
; both magnitude and direction are needed to define it. The
scalar absolute value (
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 system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Interna ...
) as
metres per second
The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre in a ...
(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, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
''.
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
![Kinematics](https://upload.wikimedia.org/wikipedia/commons/9/98/Kinematics.svg)
Speed, the
scalar 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:
:
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
slope of the tangent line to the curve at any point, and the average velocity as the slope of the
secant line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to:
* a secant line, in geometry
* the secant variety, in algebraic geometry
* secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...
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:
:
where we may identify
:
and
:
Instantaneous velocity
![Velocity vs time graph](https://upload.wikimedia.org/wikipedia/commons/c/cf/Velocity_vs_time_graph.svg)
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 (input value). Derivatives are a fundamental tool of calculus. ...
of the position with respect to time:
:
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, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
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).
:
Since the derivative of the position with respect to time gives the change in position (in
metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its prefi ...
s) divided by the change in time (in
seconds), velocity is measured in
metres per second
The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre in a ...
(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, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
. 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 is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the
line tangent to the curve of a graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time:
:
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:
:
Constant acceleration
In the special case of constant acceleration, velocity can be studied using the
suvat equations. By considering a as being equal to some arbitrary constant vector, it is trivial to show that
:
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
:
It is also possible to derive an expression for the velocity independent of time, known as the
Torricelli equation
In physics, Torricelli's equation, or Torricelli's formula, is an equation created by Evangelista Torricelli to find the final velocity of an object moving with a constant acceleration along an axis (for example, the x axis) without having a kno ...
, as follows:
:
:
:
where etc.
The above equations are valid for both
Newtonian mechanics and
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 ...
. 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, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
of a moving object is dependent on its velocity and is given by the equation
:
ignoring
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 ...
, 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, is a vector and defined by
:
In
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 ...
, 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 relativit ...
appears frequently, and is given by
:
where γ is the Lorentz factor and ''c'' is the speed of light.
Escape velocity
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
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
Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
(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
:
where ''G'' is the
gravitational constant 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 drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
. 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, 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 ...
in which velocities depend on the choice of reference frame.
If an object A is moving with velocity
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
''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:
:
Similarly, the relative velocity of object B moving with velocity ''w'', relative to object A moving with velocity ''v'' is:
:
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:
:, if the two objects are moving in opposite directions, or:
:, if the two objects are moving in the same direction.
Polar coordinates
In polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as ''velocity made good''), and an angular velocity, 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
* Transverse force (or ''Euler force''), the tangen ...
velocity is the component of velocity along a circle centered at the origin.
:
where
* is the transverse velocity
* 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'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
of the velocity vector and the unit vector in the direction of the displacement.
:
where is displacement.
The ''magnitude of the transverse velocity'' is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also the product of the angular speed
Angular may refer to:
Anatomy
* Angular artery, the terminal part of the facial artery
* Angular bone, a large bone in the lower jaw of amphibians and reptiles
* Angular incisure, a small anatomical notch on the stomach
* Angular gyrus, a regio ...
and the magnitude of the displacement.
:
such that
:
Angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
in scalar form is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity.
:
where
* is mass
*
The expression 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
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
, 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.
See also
Notes
References
* Robert Resnick and Jearl Walker, ''Fundamentals of Physics'', Wiley; 7 Sub edition (June 16, 2004). .
External links
Velocity and Acceleration
( Carnegie Mellon University)
{{Authority control
Motion (physics)
Kinematics
Temporal rates
SI derived units