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Velocity | |
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As a change of direction occurs while the racing cars turn on the curved track, their velocity is not constant. | |

Common symbols | v, v, v→ |

Other units | mph, ft/s |

In SI base units | m/s |

Dimension | LT^{−1} |

Part of a series of articles about |

Classical mechanics |
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The **velocity** of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction of motion (e.g. 60 km/h to the north). Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

- Constant velocity vs acceleration
- Difference between speed and velocity
- Equation of motion
- Average velocity
- Instantaneous velocity
- Relationship to acceleration
- Quantities that are dependent on velocity
- Relative velocity
- Scalar velocities
- Polar coordinates
- See also
- Notes
- References
- External links

Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The scalar absolute value (magnitude) of velocity is called *speed*, being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s) or as the SI base unit of (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 has a changing velocity and is said to be undergoing an * 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.

Speed, the scalar magnitude of a velocity vector, denotes only how fast an object is moving.^{ [1] }^{ [2] }

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, **v**(*t*), over some time period Δ*t*. 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 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

If we consider * v* as velocity and

From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (* v* vs.

Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (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.

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. As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a * v*(

From there, we can obtain an expression for velocity as the area under an * a*(

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 * v* as the velocity at time

- .

It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows:

where *v* = |* v*| etc.

The above equations are valid for both Newtonian mechanics and special relativity. 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.

The kinetic energy of a moving object is dependent on its velocity and is given by the equation

ignoring special relativity, 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, the dimensionless Lorentz factor appears frequently, and is given by

where γ is the Lorentz factor and *c* is the speed of light.

Escape velocity 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

where *G* is the Gravitational constant and *g* is the Gravitational acceleration. 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** 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 which velocities depend on the choice of reference frame.

If an object A is moving with velocity vector * v* and an object B with velocity vector

Similarly, the relative velocity of object B moving with velocity * w*, relative to object A moving with velocity

Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest.

In the one-dimensional case,^{ [3] } 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.

In polar coordinates, 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 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 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 and the magnitude of the displacement.

such that

Angular momentum 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, 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.

- Four-velocity (relativistic version of velocity for Minkowski spacetime)
- Group velocity
- Hypervelocity
- Phase velocity
- Proper velocity (in relativity, using traveler time instead of observer time)
- Rapidity (a version of velocity additive at relativistic speeds)
- Terminal velocity
- Velocity vs. time graph

- ↑ Rowland, Todd (2019). "Velocity Vector". Wolfram MathWorld. Retrieved 2 June 2019.
- ↑ Wilson, Edwin Bidwell (1901).
*Vector analysis: a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs*. p. 125. - ↑ Basic principle

In mechanics, **acceleration** is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities. The orientation of an object's acceleration is given by the orientation of the **net** force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law., is the combined effect of two causes:

In physics, **angular momentum** is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.

**Torque**, **moment**, **moment of force** or "turning effect" is the rotational equivalent of linear force. The concept originated with the studies by Archimedes of the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action of force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter *tau*. When being referred to as moment of force, it is commonly denoted by *M*.

In physics, **equations of motion** are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

**Kinematics** is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In physics, **angular velocity** refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. In general, angular velocity is measured in angle per unit time, e.g. radians per second. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as *1/sec*. Angular velocity is usually represented by the symbol omega. By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.

**Work** is the product of force and displacement. In physics, a force is said to do work if, when acting, there is a movement of the point of application in the direction of the force.

The **moment of inertia**, otherwise known as the **angular mass** or **rotational inertia**, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation rate. It is an extensive (additive) property: for a point mass the moment of inertia is just the mass times the square of the perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems. Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

**Rigid-body dynamics** studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

In physics, **circular motion** is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

A **fictitious force** is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as an accelerating or rotating reference frame. An example is seen in a passenger vehicle that is accelerating in the forward direction - passengers perceive that they are acted upon by a force in the rearward direction pushing them back into their seats. An example in a rotating reference frame is the force that appears to push objects outwards towards the rim of a centrifuge. These apparent forces are examples of fictitious forces.

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the **particle acceleration** or sound acceleration with the symbol a in metre/second^{2}. In acoustics or physics, **acceleration** is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time^{2}. In SI units, this is m/s^{2}.

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 particle following a curvilinear orbit to the angular velocity of the orbital motion.

**Rotation around a fixed axis** or **about a fixed axis of revolution** or **motion with respect to a fixed axis of rotation** is a special case of rotational motion. The fixed axis hypothesis excludes the possibility of an axis changing its orientation, and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear.

The **Liénard–Wiechert potentials** describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Built directly from Maxwell's equations, these potentials describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum-mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900.

**Linear motion** is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle along a line can be described by its position , which varies with (time). An example of linear motion is an athlete running 100m along a straight track.

**Classical mechanics** describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

In physics, **relativistic angular momentum** refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics.

- Robert Resnick and Jearl Walker,
*Fundamentals of Physics*, Wiley; 7 Sub edition (June 16, 2004). ISBN 0-471-23231-9.

Wikimedia Commons has media related to . Velocity |

- physicsabout.com, Speed and Velocity
- Velocity and Acceleration
- Introduction to Mechanisms (Carnegie Mellon University)

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