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Velocity

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From Wikipedia, the free encyclopedia
Speed and direction of a motion
This article is about velocity in physics. For other uses, seeVelocity (disambiguation).

Velocity
As a change of direction occurs while the racing cars turn on the curved track, their velocity is not constant even if their speed is.
Common symbols
v,v,v→,v
Other units
mph,ft/s
InSI base unitsm/s
DimensionLT−1
Part of a series on
Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}

Velocity is thespeed in combination with the direction ofmotion of anobject. Velocity is a fundamental concept inkinematics, the branch ofclassical mechanics that describes the motion of bodies.

Velocity is a physicalvectorquantity: both magnitude and direction are needed to define it. Thescalarabsolute value (magnitude) of velocity is calledspeed, being a coherent derived unit whose quantity is measured in theSI (metric system) asmetres 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 anacceleration.

Definition

Average velocity

Theaverage velocity of an object over a period of time is itschange in position,Δs{\displaystyle \Delta s}, divided by the duration of the period,Δt{\displaystyle \Delta t}, given mathematically as[1]v¯=ΔsΔt.{\displaystyle {\bar {v}}={\frac {\Delta s}{\Delta t}}.}

Instantaneous velocity

Example of a velocity vs. time graph, and the relationship between velocityv on the y-axis, accelerationa (the three greentangent lines represent the values for acceleration at different points along the curve) and displacements (the yellowarea under the curve.)

Theinstantaneousvelocity of an object is the limit average velocity as the time interval approaches zero. At any particular timet, it can be calculated as thederivative of the position with respect to time:[2]v=limΔt→0ΔsΔt=dsdt.{\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s}}}{\Delta t}}={\frac {d{\boldsymbol {s}}}{dt}}.}

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,s. In calculus terms, theintegral of the velocity functionv(t) is the displacement functions(t). In the figure, this corresponds to the yellow area under the curve.s=∫v dt.{\displaystyle {\boldsymbol {s}}=\int {\boldsymbol {v}}\ dt.}

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.

Difference between speed and velocity

Main article:Speed
Kinematic quantities of a classical particle: massm, positionr, velocityv, accelerationa.

While the termsspeed andvelocity are often colloquially used interchangeably to connote how fast an object is moving, in scientific terms they are different. Speed, thescalar 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 aconstant 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.

Units

Since the derivative of the position with respect to time gives the change in position (inmetres) divided by the change in time (inseconds), velocity is measured inmetres per second (m/s).

Equation of motion

Main article: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 theinstantaneous 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 periodΔt. Average velocity can be calculated as:[6][7]

v¯=ΔxΔt=∫t0t1v(t)dtt1−t0.{\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {x} }{\Delta t}}={\frac {\int _{t_{0}}^{t_{1}}\mathbf {v} (t)dt}{t_{1}-t_{0}}}.}

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 theslope of the tangent line to the curve at any point, and the average velocity as the slope of thesecant line between two points witht coordinates equal to the boundaries of the time period for the average velocity.

Special cases

Ift1 =t2 =t3 = ... =t, then average speed is given by thearithmetic mean of the speedsv¯=v1+v2+v3+⋯+vnn=1n∑i=1nvi{\displaystyle {\bar {v}}={v_{1}+v_{2}+v_{3}+\dots +v_{n} \over n}={\frac {1}{n}}\sum _{i=1}^{n}{v_{i}}}

  • When a particle moves different distancess1,s2,s3,...,sn with speedsv1,v2,v3,...,vn respectively, then the average speed of the particle over the total distance is given as[8]

v¯=s1+s2+s3+⋯+snt1+t2+t3+⋯+tn=s1+s2+s3+⋯+sns1v1+s2v2+s3v3+⋯+snvn{\displaystyle {\bar {v}}={s_{1}+s_{2}+s_{3}+\dots +s_{n} \over t_{1}+t_{2}+t_{3}+\dots +t_{n}}={{s_{1}+s_{2}+s_{3}+\dots +s_{n}} \over {{s_{1} \over v_{1}}+{s_{2} \over v_{2}}+{s_{3} \over v_{3}}+\dots +{s_{n} \over v_{n}}}}}Ifs1 =s2 =s3 = ... =s, then average speed is given by theharmonic mean of the speeds[8]v¯=n(1v1+1v2+1v3+⋯+1vn)−1=n(∑i=1n1vi)−1.{\displaystyle {\bar {v}}=n\left({1 \over v_{1}}+{1 \over v_{2}}+{1 \over v_{3}}+\dots +{1 \over v_{n}}\right)^{-1}=n\left(\sum _{i=1}^{n}{\frac {1}{v_{i}}}\right)^{-1}.}

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'sacceleration. As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at apoint in time is theslope of theline tangent to the curve of av(t) graph at that point. In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time:[9]a=dvdt.{\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}.}

From there, velocity is expressed as the area under ana(t) acceleration vs. time graph. As above, this is done using the concept of the integral:

v=∫a dt.{\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.}

Constant acceleration

In the special case of constant acceleration, velocity can be studied using thesuvat equations. By consideringa as being equal to some arbitrary constant vector, this showsv=u+at{\displaystyle {\boldsymbol {v}}={\boldsymbol {u}}+{\boldsymbol {a}}t}withv as the velocity at timet andu as the velocity at timet = 0. By combining this equation with the suvat equationx =ut +at2/2, it is possible to relate the displacement and the average velocity byx=(u+v)2t=v¯t.{\displaystyle {\boldsymbol {x}}={\frac {({\boldsymbol {u}}+{\boldsymbol {v}})}{2}}t={\boldsymbol {\bar {v}}}t.}It is also possible to derive an expression for the velocity independent of time, known as theTorricelli equation, as follows:v2=v⋅v=(u+at)⋅(u+at)=u2+2t(a⋅u)+a2t2{\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}}(2a)⋅x=(2a)⋅(ut+12at2)=2t(a⋅u)+a2t2=v2−u2{\displaystyle (2{\boldsymbol {a}})\cdot {\boldsymbol {x}}=(2{\boldsymbol {a}})\cdot ({\boldsymbol {u}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2})=2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}=v^{2}-u^{2}}∴v2=u2+2(a⋅x){\displaystyle \therefore v^{2}=u^{2}+2({\boldsymbol {a}}\cdot {\boldsymbol {x}})}wherev = |v| etc.

The above equations are valid for bothNewtonian mechanics andspecial 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.

Quantities that are dependent on velocity

Momentum

In classical mechanics,Newton's second law definesmomentum, p, as a vector that is the product of an object's mass and velocity, given mathematically asp=mv{\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}}wherem is the mass of the object.

Kinetic energy

Thekinetic energy of a moving object is dependent on its velocity and is given by the equation[10]Ek=12mv2{\displaystyle E_{\text{k}}={\tfrac {1}{2}}mv^{2}}whereEk is the kinetic energy. Kinetic energy is a scalar quantity as it depends on the square of the velocity.

Drag (fluid resistance)

Influid dynamics,drag is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force,FD{\displaystyle F_{D}}, is dependent on the square of velocity and is given asFD=12ρv2CDA{\displaystyle F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,A}where

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'sgravitational potential energy (which is always negative), is equal to zero. The general formula for the escape velocity of an object at a distancer from the center of a planet with massM is[12]ve=2GMr=2gr,{\displaystyle v_{\text{e}}={\sqrt {\frac {2GM}{r}}}={\sqrt {2gr}},}whereG is thegravitational constant andg is thegravitational 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 does not intersect with something in its path.

The Lorentz factor of special relativity

Inspecial relativity, the dimensionlessLorentz factor appears frequently, and is given by[13]γ=11−v2c2{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}where γ is the Lorentz factor andc is the speed of light.

Relative velocity

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 velocityvectorv and an object B with velocity vectorw; theseabsolute velocities are typically expressed in the sameinertial reference frame. Then, the velocity of object Arelative to object B is defined as the difference of the two velocity vectors:vA 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 velocityw, relative to object A moving with velocityv is:vB relative to A=w−v{\displaystyle {\boldsymbol {v}}_{B{\text{ relative to }}A}={\boldsymbol {w}}-{\boldsymbol {v}}}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 withspecial relativity in which velocities depend on the choice of reference frame.

Scalar velocities

In the one-dimensional case,[14] the velocities are scalars and the equation is either:vrel=v−(−w),{\displaystyle v_{\text{rel}}=v-(-w),} if the two objects are moving in opposite directions, or:vrel=v−(+w),{\displaystyle v_{\text{rel}}=v-(+w),} if the two objects are moving in the same direction.

Coordinate systems

Cartesian coordinates

In multi-dimensionalCartesian 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]

vx=dx/dt,{\displaystyle v_{x}=dx/dt,}

vy=dy/dt.{\displaystyle v_{y}=dy/dt.}

The two-dimensional velocity vector is then defined asv=<vx,vy>{\displaystyle {\textbf {v}}=<v_{x},v_{y}>}. The magnitude of this vector represents speed and is found by thedistance formula as

|v|=vx2+vy2.{\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}}}.}

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

vz=dz/dt.{\displaystyle v_{z}=dz/dt.}

The three-dimensional velocity vector is defined asv=<vx,vy,vz>{\displaystyle {\textbf {v}}=<v_{x},v_{y},v_{z}>} with its magnitude also representing speed and being determined by

|v|=vx2+vy2+vz2.{\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}.}

While some textbooks use subscript notation to define Cartesian components of velocity, others useu{\displaystyle u},v{\displaystyle v}, andw{\displaystyle w} for thex{\displaystyle x}-,y{\displaystyle y}-, andz{\displaystyle z}-axes respectively.[16]

Polar coordinates

See also:Circular_motion § In_polar_coordinates; andRadial, transverse, normal
Representation of radial and tangential components of velocity at different moments of linear motion with constant velocity of the object around an observer O (it corresponds, for example, to the passage of a car on a straight street around a pedestrian standing on the sidewalk). The radial component can be observed due to theDoppler effect, the tangential component causes visible changes of the position of the object.

Inpolar coordinates, a two-dimensional velocity is described by aradial velocity, defined as the component of velocity away from or toward the origin, and atransverse velocity, perpendicular to the radial one.[17][18] Both arise fromangular 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. Thetransverse velocity is the component of velocity along a circle centered at the origin.v=vT+vR{\displaystyle {\boldsymbol {v}}={\boldsymbol {v}}_{T}+{\boldsymbol {v}}_{R}}where

Theradial speed (or magnitude of the radial velocity) is thedot product of the velocity vector and the unit vector in the radial direction.vR=v⋅r|r|=v⋅r^{\displaystyle v_{R}={\frac {{\boldsymbol {v}}\cdot {\boldsymbol {r}}}{\left|{\boldsymbol {r}}\right|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {r}}}}wherer{\displaystyle {\boldsymbol {r}}} is position andr^{\displaystyle {\hat {\boldsymbol {r}}}} is the radial direction.

The transverse speed (or magnitude of the transverse velocity) is the magnitude of thecross 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 theangular speedω{\displaystyle \omega } and the radius (the magnitude of the position).vT=|r×v||r|=v⋅t^=ω|r|{\displaystyle v_{T}={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {t}}}=\omega |{\boldsymbol {r}}|}such thatω=|r×v||r|2.{\displaystyle \omega ={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|^{2}}}.}

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=mrvT=mr2ω{\displaystyle L=mrv_{T}=mr^{2}\omega }where

The expressionmr2{\displaystyle mr^{2}} is known asmoment of inertia.If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitationalorbit, 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 asKepler's laws of planetary motion.

See also

Notes

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

References

  1. ^"The Feynman Lectures on Physics Vol. I Ch. 8: Motion".www.feynmanlectures.caltech.edu. Retrieved2024-01-05.
  2. ^David Halliday; Robert Resnick; Jearl Walker (2021).Fundamentals of Physics, Extended (12th ed.). John Wiley & Sons. p. 71.ISBN 978-1-119-77351-1.Extract of page 71
  3. ^Richard P. Olenick; Tom M. Apostol; David L. Goodstein (2008).The Mechanical Universe: Introduction to Mechanics and Heat (illustrated, reprinted ed.). Cambridge University Press. p. 84.ISBN 978-0-521-71592-8.Extract of page 84
  4. ^Michael J. Cardamone (2007).Fundamental Concepts of Physics. Universal-Publishers. p. 5.ISBN 978-1-59942-433-0.Extract of page 5
  5. ^Jerry D. Wilson; Anthony J. Buffa; Bo Lou (2022).College Physics Essentials, Eighth Edition (Two-Volume Set) (illustrated ed.). CRC Press. p. 40.ISBN 978-1-351-12991-6.Extract of page 40
  6. ^David Halliday; Robert Resnick; Jearl Walker (2021).Fundamentals of Physics, Extended (12th ed.). John Wiley & Sons. p. 70.ISBN 978-1-119-77351-1.Extract of page 70
  7. ^Adrian Banner (2007).The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (illustrated ed.). Princeton University Press. p. 350.ISBN 978-0-691-13088-0.Extract of page 350
  8. ^abGiri & Bannerjee (2002).Statistical Tools and Technique. Academic Publishers. p. 4.ISBN 978-81-87504-39-9.Extract of page 4
  9. ^Bekir Karaoglu (2020).Classical Physics: A Two-Semester Coursebook. Springer Nature. p. 41.ISBN 978-3-030-38456-2.Extract of page 41
  10. ^David Halliday; Robert Resnick; Jearl Walker (2010).Fundamentals of Physics, Chapters 33-37. John Wiley & Sons. p. 1080.ISBN 978-0-470-54794-6.Extract of page 1080
  11. ^ForEarth's atmosphere, the air density can be found using thebarometric formula. It is 1.293 kg/m3 at 0 °C and 1atmosphere.
  12. ^Jim Breithaupt (2000).New Understanding Physics for Advanced Level (illustrated ed.). Nelson Thornes. p. 231.ISBN 978-0-7487-4314-8.Extract of page 231
  13. ^Eckehard W Mielke (2022).Modern Aspects Of Relativity. World Scientific. p. 98.ISBN 978-981-12-4406-3.Extract of page 98
  14. ^Basic principle
  15. ^"The Feynman Lectures on Physics Vol. I Ch. 9: Newton's Laws of Dynamics".www.feynmanlectures.caltech.edu. Retrieved2024-01-04.
  16. ^White, F. M. (2008).Fluid mechanics. The McGraw Hill Companies,.
  17. ^E. Graham; Aidan Burrows; Brian Gaulter (2002).Mechanics, Volume 6 (illustrated ed.). Heinemann. p. 77.ISBN 978-0-435-51311-5.Extract of page 77
  18. ^Anup Goel; H. J. Sawant (2021).Engineering Mechanics. Technical Publications. p. 8.ISBN 978-93-332-2190-0.Extract of page 8

External links

Wikimedia Commons has media related toVelocity.
Linear/translational quantitiesAngular/rotational quantities
Dimensions1LL2Dimensions1θθ2
Ttime:t
s		absement:A
m s		Ttime:t
s
1distance:d,position:r,s,x,displacement
m		area:A
m2		1angle:θ,angular displacement:θ
rad		solid angle:Ω
rad2, sr
T−1frequency:f
s−1,Hz		speed:v,velocity:v
m s−1		kinematic viscosity:ν,
specific angular momentum: h
m2 s−1		T−1frequency:f,rotational speed:n,rotational velocity:n
s−1,Hz		angular speed:ω,angular velocity:ω
rad s−1
T−2acceleration:a
m s−2		T−2rotational acceleration
s−2		angular acceleration:α
rad s−2
T−3jerk:j
m s−3		T−3angular jerk:ζ
rad s−3
Mmass:m
kg		weighted position:M ⟨x⟩ = ∑mxmoment of inertia: I
kg m2		ML
MT−1Mass flow rate:m˙{\displaystyle {\dot {m}}}
kg s−1		momentum:p,impulse:J
kg m s−1,N s		action:𝒮,actergy:ℵ
kg m2 s−1,J s		MLT−1angular momentum:L,angular impulse:ΔL
kg m rad s−1
MT−2force:F,weight:Fg
kg m s−2,N		energy:E,work:W,Lagrangian:L
kg m2 s−2,J		MLT−2torque:τ,moment:M
kg m rad s−2,N m
MT−3yank:Y
kg m s−3, N s−1		power:P
kg m2 s−3, W		MLT−3rotatum:P
kg m rad s−3, N m s−1
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