Velocity | |
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Common symbols | v, v, v→, v |

Other units | mph, ft/s |

In SI base units | m/s |

Dimension | L T^{−1} |

Part of a series on |

Classical mechanics |
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**Velocity** is the speed in combination with the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

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 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*.

The **average velocity** of an object over a period of time is its change in position, , divided by the duration of the period, , given mathematically as^{[1]}

The **instantaneous** **velocity** of an object is the limit average velocity as the time interval approaches zero. At any particular time *t*, it can be calculated as the derivative of the position with respect to time:^{[2]}

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

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.

Main article: Speed |

While the terms *speed* and *velocity* are often colloquially used interchangeably to connote how fast an object is moving, in scientific terms they are different. Speed, the scalar magnitude of a velocity vector, denotes only how fast an object is moving, while velocity indicates both an object's speed and direction.^{[3]}^{[4]}^{[5]}

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.

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).

Main article: Equation of motion |

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:^{[6]}^{[7]}

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.

- When a particle moves with different uniform speeds
*v*_{1},*v*_{2},*v*_{3}, ...,*v*_{n}in different time intervals*t*_{1},*t*_{2},*t*_{3}, ...,*t*_{n}respectively, then average speed over the total time of journey is given as

If *t*_{1} = *t*_{2} = *t*_{3} = ... = *t*, then average speed is given by the arithmetic mean of the speeds

- When a particle moves different distances
*s*_{1},*s*_{2},*s*_{3},...,*s*_{n}with speeds*v*_{1},*v*_{2},*v*_{3},...,*v*_{n}respectively, then the average speed of the particle over the total distance is given as^{[8]}

If

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

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

where

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.

In classical mechanics, Newton's second law defines momentum, p, as a vector that is the product of an object's mass and velocity, given mathematically as

whereThe kinetic energy of a moving object is dependent on its velocity and is given by the equation^{[10]}

In fluid dynamics, drag is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force, , is dependent on the square of velocity and is given as

where

- is the density of the fluid,
^{[11]} - is the speed of the object relative to the fluid,
- is the cross sectional area, and
- is the drag coefficient – a dimensionless number.

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^{[12]}

where

In special relativity, the dimensionless Lorentz factor appears frequently, and is given by^{[13]}

Main article: 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.

Consider an object A moving with velocity vector * v* and an object B with velocity vector

Similarly, the relative velocity of object B moving with velocity

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

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.

In the one-dimensional case,^{[14]} 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 multi-dimensional Cartesian coordinate systems, velocity is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as^{[15]}

The two-dimensional velocity vector is then defined as . The magnitude of this vector represents speed and is found by the distance formula as

In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as

The three-dimensional velocity vector is defined as with its magnitude also representing speed and being determined by

While some textbooks use subscript notation to define Cartesian components of velocity, others use , , and for the -, -, and -axes respectively.^{[16]}

See also: Circular_motion § In_polar_coordinates; and Radial, transverse, normal |

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, and a *transverse velocity*, perpendicular to the radial one.^{[17]}^{[18]} Both arise from 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 traverse 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 *radial speed* (or magnitude of the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction.

where is position and is the radial direction.

The transverse speed (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector. It is also the dot product of velocity and transverse direction, or the product of the angular speed and the radius (the magnitude of the position).

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.