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Acceleration

From Wikipedia, the free encyclopedia
Rate of change of velocity
This article is about acceleration in physics. For other uses, seeAcceleration (disambiguation).
"Accelerate" redirects here. For other uses, seeAccelerate (disambiguation).

Acceleration
In vacuum (noair resistance), objects attracted by Earth gain speed at a steady rate.
Common symbols
a
SI unitm/s2, m·s−2, m s−2
Derivations from
other quantities
a=dvdt=d2xdt2{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}}
DimensionLT−2{\displaystyle {\mathsf {L}}{\mathsf {T}}^{-2}}
Part of a series on
Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
Drag racing is a sport in which specially-built vehicles compete to be the fastest to accelerate from a standing start.

Inmechanics,acceleration is therate of change of thevelocity of an object with respect to time. Acceleration is one of several components ofkinematics, the study ofmotion. Accelerations arevector quantities (in that they havemagnitude anddirection).[1][2] The orientation of an object's acceleration is given by the orientation of thenetforce acting on that object. The magnitude of an object's acceleration, as described byNewton's second law,[3] is the combined effect of two causes:

TheSI unit for acceleration ismetre per second squared (m⋅s−2,ms2{\displaystyle \mathrm {\tfrac {m}{s^{2}}} }).

For example, when avehicle starts from astandstill (zero velocity, in aninertial frame of reference) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the vehicle turns, an acceleration occurs toward the new direction and changes its motion vector. The acceleration of the vehicle in its current direction of motion is called alinear acceleration (ortangential acceleration duringcircular motions), thereaction to which the passengers on board experience as a force pushing them back into their seats. When changing direction, the effecting acceleration is calledradial ornormal acceleration (orcentripetal acceleration during circular motions), the reaction to which the passengers experience as acentrifugal force. If the speed of the vehicle decreases, this is an acceleration in the opposite direction of the velocity vector, sometimes calleddeceleration[4][5] orretardation, and passengers experience the reaction to deceleration as aninertial force pushing them forward. Such deceleration is often achieved byretrorocket burning inspacecraft.[6] Both acceleration and deceleration are treated the same, as they are both changes in velocity. Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their relative (differential) velocity is neutralised inreference to the acceleration due to change in speed.

Definition and properties

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

Average acceleration

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Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at timet is found in the limit astime intervalΔt → 0 ofΔv/Δt.

An object's average acceleration over a period oftime is its change invelocity,Δv{\displaystyle \Delta \mathbf {v} }, divided by the duration of the period,Δt{\displaystyle \Delta t}. Mathematically,a¯=ΔvΔt.{\displaystyle {\bar {\mathbf {a} }}={\frac {\Delta \mathbf {v} }{\Delta t}}.}

Instantaneous acceleration

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From bottom to top:
  • an acceleration functiona(t);
  • the integral of the acceleration is the velocity functionv(t);
  • and the integral of the velocity is the distance functions(t).

Instantaneous acceleration, meanwhile, is thelimit of the average acceleration over aninfinitesimal interval of time. In the terms ofcalculus, instantaneous acceleration is thederivative of the velocity vector with respect to time:a=limΔt→0ΔvΔt=dvdt.{\displaystyle \mathbf {a} =\lim _{{\Delta t}\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {d\mathbf {v} }{dt}}.}As acceleration is defined as the derivative of velocity,v, with respect to timet and velocity is defined as the derivative of position,x, with respect to time, acceleration can be thought of as thesecond derivative ofx with respect tot:a=dvdt=d2xdt2.{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}.}

(Here and elsewhere, ifmotion is in a straight line,vector quantities can be substituted byscalars in the equations.)

By thefundamental theorem of calculus, it can be seen that theintegral of the acceleration functiona(t) is the velocity functionv(t); that is, the area under the curve of an acceleration vs. time (a vs.t) graph corresponds to the change of velocity.Δv=∫adt.{\displaystyle \Delta \mathbf {v} =\int \mathbf {a} \,dt.}

Likewise, the integral of thejerk functionj(t), the derivative of the acceleration function, can be used to find the change of acceleration at a certain time:Δa=∫jdt.{\displaystyle \Delta \mathbf {a} =\int \mathbf {j} \,dt.}

Units

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Acceleration has thedimensions of velocity (L/T) divided by time, i.e.LT−2. TheSI unit of acceleration is themetre per second squared (m s−2); or "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.

Other forms

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An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, although its speed may be constant. In this case it is said to be undergoingcentripetal (directed towards the center) acceleration.

Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called anaccelerometer.

Inclassical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the netforce vector (i.e. sum of all forces) acting on it (Newton's second law):F=ma⟹a=Fm,{\displaystyle \mathbf {F} =m\mathbf {a} \quad \implies \quad \mathbf {a} ={\frac {\mathbf {F} }{m}},}whereF is the net force acting on the body,m is themass of the body, anda is the center-of-mass acceleration. As speeds approach thespeed of light,relativistic effects become increasingly large.

Tangential and centripetal acceleration

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See also:Centripetal force § Local coordinates, andTangential velocity
An oscillating pendulum, with velocity and acceleration marked. It experiences both tangential and centripetal acceleration.
Components of acceleration for a curved motion. The tangential componentat is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion)ac is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.

The velocity of a particle moving on a curved path as afunction of time can be written as:v=vvv=vut,{\displaystyle \mathbf {v} =v{\frac {\mathbf {v} }{v}}=v\mathbf {u} _{\mathrm {t} },}withv equal to the speed of travel along the path, andut=vv,{\displaystyle \mathbf {u} _{\mathrm {t} }={\frac {\mathbf {v} }{v}}\,,}aunit vector tangent to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speedv and the changing direction ofut, the acceleration of a particle moving on a curved path can be written using thechain rule of differentiation[7] for the product of two functions of time as:

a=dvdt=dvdtut+vdutdt=dvdtut+v2run ,{\displaystyle {\begin{alignedat}{3}\mathbf {a} &={\frac {d\mathbf {v} }{dt}}\\&={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }+v{\frac {d\mathbf {u} _{\mathrm {t} }}{dt}}\\&={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }+{\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }\ ,\end{alignedat}}}

whereun is the unit (inward)normal vector to the particle's trajectory (also calledthe principal normal), andr is its instantaneousradius of curvature based upon theosculating circle at timet. The componentsat=dvdtutandac=v2run{\displaystyle \mathbf {a} _{\mathrm {t} }={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }\quad {\text{and}}\quad \mathbf {a} _{\mathrm {c} }={\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }}are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see alsocircular motion andcentripetal force), respectively.

Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by theFrenet–Serret formulas.[8][9]

Special cases

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Uniform acceleration

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See also:Torricelli's equation
Calculation of the speed difference for a uniform acceleration

Uniform orconstant acceleration is a type of motion in which thevelocity of an object changes by an equal amount in every equal time period.

A frequently cited example of uniform acceleration is that of an object infree fall in a uniform gravitational field. The acceleration of a falling body in the absence of resistances to motion is dependent only on thegravitational field strengthg (also calledacceleration due to gravity). ByNewton's second law theforceFg{\displaystyle \mathbf {F_{g}} } acting on a body is given by:Fg=mg.{\displaystyle \mathbf {F_{g}} =m\mathbf {g} .}

Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating thedisplacement, initial and time-dependentvelocities, and acceleration to thetime elapsed:[10]s(t)=s0+v0t+12at2=s0+12(v0+v(t))tv(t)=v0+atv2(t)=v02+2a⋅[s(t)−s0],{\displaystyle {\begin{aligned}\mathbf {s} (t)&=\mathbf {s} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}=\mathbf {s} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} _{0}+\mathbf {v} (t)\right)t\\\mathbf {v} (t)&=\mathbf {v} _{0}+\mathbf {a} t\\{v^{2}}(t)&={v_{0}}^{2}+2\mathbf {a\cdot } [\mathbf {s} (t)-\mathbf {s} _{0}],\end{aligned}}}

where

In particular, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations. AsGalileo showed, the net result is parabolic motion, which describes, e.g., the trajectory of a projectile in vacuum near the surface of Earth.[11]

Circular motion

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Position vectorr, always points radially from the origin.
Velocity vectorv, always tangent to the path of motion.
Acceleration vectora, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
Kinematic vectors in planepolar coordinates. Notice the setup is not restricted to 2d space, but may represent theosculating plane plane in a point of an arbitrary curve in any higher dimension.

In uniformcircular motion, that is moving with constantspeed along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant. The derivative of the location of a point on a curve with respect to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to the radius in this point. Since in uniform motion the velocity in the tangential direction does not change, the acceleration must be in radial direction, pointing to the center of the circle. This acceleration constantly changes the direction of the velocity to be tangent in the neighbouring point, thereby rotating the velocity vector along the circle.

Expressing centripetal acceleration vector in polar components, wherer{\displaystyle \mathbf {r} } is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yieldsac=−v2|r|⋅r|r|.{\displaystyle \mathbf {a} _{c}=-{\frac {v^{2}}{|\mathbf {r} |}}\cdot {\frac {\mathbf {r} }{|\mathbf {r} |}}\,.}

As usual in rotations, the speedv{\displaystyle v} of a particle may be expressed as anangular speed with respect to a point at the distancer{\displaystyle r} asω=vr.{\displaystyle \omega ={\frac {v}{r}}.}

Thusac=−ω2r.{\displaystyle \mathbf {a} _{c}=-\omega ^{2}\mathbf {r} \,.}

This acceleration and the mass of the particle determine the necessarycentripetal force, directedtoward the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion. The so-called 'centrifugal force', appearing to act outward on the body, is a so-calledpseudo force experienced in theframe of reference of the body in circular motion, due to the body'slinear momentum, a vector tangent to the circle of motion.

In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to theprincipal normal, which directs to the center of the osculating circle, that determines the radiusr{\displaystyle r} for the centripetal acceleration. The tangential component is given by the angular accelerationα{\displaystyle \alpha }, i.e., the rate of changeα=ω˙{\displaystyle \alpha ={\dot {\omega }}} of the angular speedω{\displaystyle \omega } times the radiusr{\displaystyle r}. That is,at=rα.{\displaystyle a_{t}=r\alpha .}

The sign of the tangential component of the acceleration is determined by the sign of theangular acceleration (α{\displaystyle \alpha }), and the tangent is always directed at right angles to the radius vector.

Coordinate systems

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In multi-dimensionalCartesian coordinate systems, acceleration 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 acceleration components are defined as[12]ax=dvxdt=d2xdt2,ay=dvydt=d2ydt2.{\displaystyle {\begin{aligned}a_{x}&={\frac {dv_{x}}{dt}}={\frac {d^{2}x}{dt^{2}}},\\a_{y}&={\frac {dv_{y}}{dt}}={\frac {d^{2}y}{dt^{2}}}.\end{aligned}}}The two-dimensional acceleration vector is then defined asa=⟨ax,ay⟩{\displaystyle \mathbf {a} =\langle a_{x},a_{y}\rangle }. The magnitude of this vector is found by thedistance formula as|a|=ax2+ay2.{\displaystyle |a|={\sqrt {a_{x}^{2}+a_{y}^{2}}}.}In three-dimensional systems where there is an additional z-axis, the corresponding acceleration component is defined asaz=dvzdt=d2zdt2.{\displaystyle a_{z}={\frac {dv_{z}}{dt}}={\frac {d^{2}z}{dt^{2}}}.}The three-dimensional acceleration vector is defined asa=⟨ax,ay,az⟩{\displaystyle \mathbf {a} =\langle a_{x},a_{y},a_{z}\rangle } with its magnitude being determined by|a|=ax2+ay2+az2.{\displaystyle |a|={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}.}

Relation to relativity

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Special relativity

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Main articles:Special relativity andAcceleration (special relativity)

The special theory of relativity describes the behaviour of objects travelling relative to other objects at speeds approaching that of light in vacuum.Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations.

As speeds approach that of light, the acceleration produced by a given force decreases, becominginfinitesimally small as light speed is approached; an object with mass can approach this speedasymptotically, but never reach it.

General relativity

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Main article:General relativity

Unless the state of motion of an object is known, it is impossible to distinguish whether an observed force is due togravity or to acceleration—gravity and inertial acceleration have identical effects.Albert Einstein called this theequivalence principle, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating.[13]

Conversions

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Conversions between common units of acceleration
Base value(Gal, or cm/s2)(ft/s2)(m/s2)(Standard gravity,g0)
1 Gal, or cm/s210.03280840.011.01972×10−3
1 ft/s230.480010.3048000.0310810
1 m/s21003.2808410.101972
1g0980.66532.17409.806651

See also

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References

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  1. ^Bondi, Hermann (1980).Relativity and Common Sense. Courier Dover Publications. pp. 3.ISBN 978-0-486-24021-3.
  2. ^Lehrman, Robert L. (1998).Physics the Easy Way. Barron's Educational Series. pp. 27.ISBN 978-0-7641-0236-3.
  3. ^Crew, Henry (2008).The Principles of Mechanics. BiblioBazaar, LLC. p. 43.ISBN 978-0-559-36871-4.
  4. ^P. Smith; R. C. Smith (1991).Mechanics (2nd, illustrated, reprinted ed.). John Wiley & Sons. p. 39.ISBN 978-0-471-92737-2.Extract of page 39
  5. ^John D. Cutnell; Kenneth W. Johnson (2014).Physics, Volume One: Chapters 1-17, Volume 1 (1st0, illustrated ed.). John Wiley & Sons. p. 36.ISBN 978-1-118-83688-0.Extract of page 36
  6. ^Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2008).College Physics, Volume 10. Cengage. p. 32.ISBN 9780495386933.
  7. ^Weisstein, Eric W."Chain Rule".Wolfram MathWorld. Wolfram Research. Retrieved2 August 2016.
  8. ^Larry C. Andrews; Ronald L. Phillips (2003).Mathematical Techniques for Engineers and Scientists. SPIE Press. p. 164.ISBN 978-0-8194-4506-3.
  9. ^Ch V Ramana Murthy; NC Srinivas (2001).Applied Mathematics. New Delhi: S. Chand & Co. p. 337.ISBN 978-81-219-2082-7.
  10. ^Keith Johnson (2001).Physics for you: revised national curriculum edition for GCSE (4th ed.). Nelson Thornes. p. 135.ISBN 978-0-7487-6236-1.
  11. ^David C. Cassidy; Gerald James Holton; F. James Rutherford (2002).Understanding physics. Birkhäuser. p. 146.ISBN 978-0-387-98756-9.
  12. ^"The Feynman Lectures on Physics Vol. I Ch. 9: Newton's Laws of Dynamics".www.feynmanlectures.caltech.edu. Retrieved2024-01-04.
  13. ^Greene, Brian (8 February 2005).The Fabric of the Cosmos: Space, Time, and the Texture of Reality. Vintage. p. 67.ISBN 0-375-72720-5.

External links

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Wikimedia Commons has media related toAcceleration.
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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