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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 its speed and direction of motion (e.g. 7001600000000000000♠60 km/h to the north). Velocity
Velocity
is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity
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.

Contents

1 Constant velocity vs acceleration 2 Distinction between speed and velocity 3 Equation of motion

3.1 Average velocity 3.2 Instantaneous velocity 3.3 Relationship to acceleration

3.3.1 Constant acceleration

3.4 Quantities that are dependent on velocity

4 Relative velocity

4.1 Scalar velocities

5 Polar coordinates 6 See also 7 Notes 8 References 9 External links

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. Distinction between speed and velocity

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

Speed
Speed
describes only how fast an object is moving, whereas velocity gives both how fast it is and in which direction the object is moving.[1] If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be noticed when we consider movement around a circle. When something moves in a circular path (at a constant speed, see above) and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Equation of motion Main article: Equation of motion Average velocity 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, v(t), over some time period Δt. Average velocity can be calculated as:

v ¯

=

Δ

x

Δ

t

.

displaystyle boldsymbol bar v = frac Delta boldsymbol x Delta mathit t .

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:

v ¯

=

1

t

1

t

0

t

0

t

1

v

( t )   d t ,

displaystyle boldsymbol bar v = 1 over t_ 1 -t_ 0 int _ t_ 0 ^ t_ 1 boldsymbol v (t) dt,

where we may identify

Δ

x

=

t

0

t

1

v

( t )   d t

displaystyle Delta boldsymbol x =int _ t_ 0 ^ t_ 1 boldsymbol v (t) dt

and

Δ t =

t

1

t

0

.

displaystyle Delta t=t_ 1 -t_ 0 .

Instantaneous velocity

Example of a velocity vs. time graph, and the relationship between velocity v on the y-axis, acceleration a (the three green tangent lines represent the values for acceleration at different points along the curve) and displacement s (the yellow area under the curve.)

If we consider v as velocity and x as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time:

v

=

lim

Δ t

→ 0

Δ

x

Δ t

=

d

x

d

t

.

displaystyle boldsymbol v =lim _ Delta t to 0 frac Delta boldsymbol x Delta t = frac d boldsymbol x d mathit t .

From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, x. In calculus terms, the integral of the velocity function v(t) is the displacement function x(t). In the figure, this corresponds to the yellow area under the curve labeled s (s being an alternative notation for displacement).

x

= ∫

v

  d

t

.

displaystyle boldsymbol x =int boldsymbol v d mathit t .

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. 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. 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(t) graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time:

a

=

d

v

d

t

.

displaystyle boldsymbol a = frac d boldsymbol v d mathit t .

From there, we can obtain an expression for velocity as the area under an a(t) acceleration vs. time graph. As above, this is done using the concept of the integral:

v

= ∫

a

  d

t

.

displaystyle boldsymbol v =int boldsymbol a d mathit t .

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

v

=

u

+

a

t

displaystyle boldsymbol v = boldsymbol u + boldsymbol a t

with v as the velocity at time t and u as the velocity at time t = 0. By combining this equation with the suvat equation x = ut + at2/2, it is possible to relate the displacement and the average velocity by

x

=

(

u

+

v

)

2

t

=

v ¯

t

displaystyle boldsymbol x = frac ( boldsymbol u + boldsymbol v ) 2 mathit t = boldsymbol bar v mathit t

.

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

v

2

=

v

v

= (

u

+

a

t ) ⋅ (

u

+

a

t ) =

u

2

+ 2 t (

a

u

) +

a

2

t

2

displaystyle v^ 2 = boldsymbol v cdot boldsymbol v =( boldsymbol u + boldsymbol a t)cdot ( boldsymbol u + boldsymbol a t)=u^ 2 +2t( boldsymbol a cdot boldsymbol u )+a^ 2 t^ 2

( 2

a

) ⋅

x

= ( 2

a

) ⋅ (

u

t +

1 2

a

t

2

) = 2 t (

a

u

) +

a

2

t

2

=

v

2

u

2

displaystyle (2 boldsymbol a )cdot boldsymbol x =(2 boldsymbol a )cdot ( boldsymbol u t+ frac 1 2 boldsymbol a t^ 2 )=2t( boldsymbol a cdot boldsymbol u )+a^ 2 t^ 2 =v^ 2 -u^ 2

v

2

=

u

2

+ 2 (

a

x

)

displaystyle therefore v^ 2 =u^ 2 +2( boldsymbol a cdot boldsymbol x )

where v = v etc. The above equations are valid for both Newtonian mechanics
Newtonian mechanics
and special relativity. Where Newtonian mechanics
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 of a moving object is dependent on its velocity and is given by the equation

E

k

=

1 2

m

v

2

displaystyle E_ text k = tfrac 1 2 mv^ 2

ignoring special relativity, where Ek is the kinetic energy and m is the mass. Kinetic energy
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

p

= m

v

displaystyle boldsymbol p =m boldsymbol v

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

γ =

1

1 −

v

2

c

2

displaystyle gamma = frac 1 sqrt 1- frac v^ 2 c^ 2

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

v

e

=

2 G M

r

=

2 g r

,

displaystyle v_ text e = sqrt frac 2GM r = sqrt 2gr ,

where G is the Gravitational constant
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 Main article: Relative velocity Relative velocity
Relative velocity
is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity
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 w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors:

v

A

 relative to 

B

=

v

w

displaystyle boldsymbol v _ A text relative to B = boldsymbol v - boldsymbol w

Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is:

v

B

 relative to 

A

=

w

v

displaystyle boldsymbol v _ B text relative to A = boldsymbol w - boldsymbol v

Usually the inertial frame is chosen in which the latter of the two mentioned objects is in rest. Scalar velocities In the one-dimensional case,[2] the velocities are scalars and the equation is either:

v

r e l

= v − ( − w )

displaystyle ,v_ rel =v-(-w)

, if the two objects are moving in opposite directions, or:

v

r e l

= v − ( + w )

displaystyle ,v_ rel =v-(+w)

, if the two objects are moving in the same direction.

Polar coordinates 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.

v

=

v

T

+

v

R

displaystyle boldsymbol v = boldsymbol v _ T + boldsymbol v _ R

where

v

T

displaystyle boldsymbol v _ T

is the transverse velocity

v

R

displaystyle boldsymbol v _ R

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.

v

R

=

v

r

r

displaystyle v_ R = frac boldsymbol v cdot boldsymbol r left boldsymbol r right

where

r

displaystyle boldsymbol r

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

ω

displaystyle omega

and the magnitude of the displacement.

v

T

=

r

×

v

r

= ω

r

displaystyle v_ T = frac boldsymbol r times boldsymbol v boldsymbol r =omega boldsymbol r

such that

ω =

r

×

v

r

2

.

displaystyle omega = frac boldsymbol r times boldsymbol v boldsymbol r ^ 2 .

Angular momentum
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.

L = m r

v

T

= m

r

2

ω

displaystyle L=mrv_ T =mr^ 2 omega ,

where

m

displaystyle m,

is mass

r = ‖

r

‖ .

displaystyle r= boldsymbol r .

The expression

m

r

2

displaystyle 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. See also

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

Notes

^ 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.  This is the likely origin of the speed/velocity terminology in vector physics. ^ Basic principle

References

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

External links

Wikimedia Commons has media related to Velocity.

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

v t e

Kinematics

← Integrate … Differentiate →

Absement Displacement (Distance) Velocity
Velocity
(Speed) Acceleration Jerk Jounce

v t e

Classical mechanics
Classical mechanics
SI units

Linear/translational quantities

Angular/rotational quantities

Dimensions 1 L L2 Dimensions 1 1 1

T time: t s absement: A m s

T time: t s

1

distance: d, position: r, s, x, displacement m area: A m2 1

angle: θ, angular displacement: θ rad solid angle: Ω rad2, sr

T−1 frequency: f s−1, Hz speed: v, velocity: v m s−1 kinematic viscosity: ν, specific angular momentum: h m2 s−1 T−1 frequency: f s−1, Hz angular speed: ω, angular velocity: ω rad s−1

T−2

acceleration: a m s−2

T−2

angular acceleration: α rad s−2

T−3

jerk: j m s−3

T−3

angular jerk: ζ rad s−3

M mass: m kg

ML2 moment of inertia: I kg m2

MT−1

momentum: p, impulse: J kg m s−1, N s action: 𝒮, actergy: ℵ kg m2 s−1, J s ML2T−1

angular momentum: L, angular impulse: ΔL kg m2 s−1 action: 𝒮, actergy: ℵ kg m2 s−1, J s

MT−2

force: F, weight: Fg kg m s−2, N energy: E, work: W kg m2 s−2, J ML2T−2

torque: τ, moment: M kg m2 s−2, N m energy: E, work: W kg m2 s−2, J

MT−3

yank: Y kg m s−3, N s−1 power: P kg m2 s−3, W ML2T−3

rotatum: P kg m2 s−3, N m s−1 power: P kg m2 s

.