# Acceleration
**Acceleration** is the rate of change of velocity with time: a vector quantity equal to the derivative of velocity, measured in the SI derived unit meter per second squared (m/s²)[1](https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration) • [2](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes). Because velocity has both magnitude and direction, a change in either one, or in both, is an acceleration[1](https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration).
| Key fact | Detail |
|---|---|
| Definition | Instantaneous acceleration is the time derivative of velocity, \( a(t) = \frac{dv(t)}{dt} \); average acceleration over an interval is \( \bar{a} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_0}{t_f - t_0} \)[3](https://openstax.org/books/university-physics-volume-1/pages/3-key-equations) |
| SI unit | Meter per second squared, m/s², a derived unit expressed in SI base units as m·s⁻²[2](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes) |
| Vector nature | Acceleration points in the direction of the change in velocity, which need not be the direction of motion; when an object slows, acceleration is opposite to its motion[1](https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration) |
| Standard gravity | The standard acceleration of free fall is \( g_n = 9.80665 \) m/s²[4](https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b9) |
| Constant-acceleration kinematics | \( v = v_0 + at \), \( x = x_0 + v_0 t + \frac{1}{2}at^2 \), and \( v^2 = v_0^2 + 2a(x - x_0) \)[5](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration) |
| Circular motion | Centripetal acceleration \( a_c = v^2/r \) points toward the center; with changing speed a tangential acceleration \( a_T = \frac{d\lvert\vec{v}\rvert}{dt} \) is added, and the total is their vector sum[6](https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-and-nonuniform-circular-motion) |
| Cause | By Newton's second law, acceleration is directly proportional to the net external force and inversely proportional to mass: \( \vec{F}_{\text{net}} = m\vec{a} \)[7](https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law) |
## Definition and vector nature
Average acceleration is the change in velocity divided by the elapsed time, \( \bar{a} = \frac{\Delta v}{\Delta t} \). As the interval \( \Delta t \) shrinks toward zero, this average approaches the instantaneous acceleration, the slope of the tangent line on a graph of velocity versus time[1](https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration). In component form the acceleration vector is \( \vec{a}(t) = \frac{dv_x}{dt}\hat{i} + \frac{dv_y}{dt}\hat{j} + \frac{dv_z}{dt}\hat{k} \)[3](https://openstax.org/books/university-physics-volume-1/pages/3-key-equations).
The direction matters as much as the magnitude. Acceleration lies in the direction of the change in velocity, not necessarily the direction of travel. A subway train braking as it enters a station is accelerating in a direction opposite to its motion; everyday language calls this deceleration, but the term is not a vector and does not specify a direction in a coordinate system, so physics treatments avoid it and say the train accelerates opposite to its motion[1](https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration).
The unit m/s² reads literally as the change in velocity, in meters per second, every second[1](https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration). Other units in use include the foot per second squared (1 ft/s² = 0.3048 m/s²) and the gal (1 Gal = 0.01 m/s²), used in gravimetry[4](https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b9).
## Motion with constant acceleration
When acceleration is constant, the average and instantaneous accelerations are equal, and the motion is described by a small set of kinematic equations[5](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration):
\[ v = v_0 + at \]
\[ x = x_0 + v_0 t + \frac{1}{2}at^2 \]
\[ v^2 = v_0^2 + 2a(x - x_0) \]
with \( x_0 \) and \( v_0 \) the initial position and velocity. The average velocity over any interval is then simply \( \bar{v} = \frac{v_0 + v}{2} \)[5](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration). These equations can also be derived by integrating: since \( \frac{d}{dt}v(t) = a(t) \), integrating a constant \( a \) gives \( v(t) = v_0 + at \), and integrating again gives the position equation[8](https://openstax.org/books/university-physics-volume-1/pages/3-6-finding-velocity-and-displacement-from-acceleration).
The constant-acceleration assumption covers a great number of situations, and motions in which acceleration changes drastically, such as a car accelerating to top speed and then braking, can be treated in separate parts, each with its own constant acceleration[5](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration). The equations also show the limiting behavior of the definition: for a finite time interval, acceleration approaches zero as the initial and final velocities become equal, and it becomes infinite in the limit of zero elapsed time for a finite velocity change[5](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration).
In two or three dimensions, motion with constant acceleration separates into independent one-dimensional motions along perpendicular axes. For projectile motion in a uniform gravitational field with air resistance neglected, the components are \( a_y = -g = -9.8 \) m/s² and \( a_x = 0 \), so the horizontal velocity stays constant while the vertical velocity changes at a steady rate[9](https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion).
## Acceleration in circular motion
An object moving at constant speed in a circle is still accelerating, because its direction of travel changes continuously. This centripetal acceleration has magnitude
\[ a_c = \frac{v^2}{r} = \omega^2 r \]
where \( v \) is the speed, \( r \) the radius, and \( \omega \) the angular velocity in radians per second, and it is directed toward the center of the circular path[6](https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-and-nonuniform-circular-motion) • [10](https://cdaweb.gsfc.nasa.gov/pub/documents/archived_websites/pwg.gsfc.nasa.gov/stargaze/Lcircul.htm). Differentiating the position vector for uniform circular motion, \( \vec{r}(t) = A\cos\omega t\,\hat{i} + A\sin\omega t\,\hat{j} \), gives an acceleration \( \vec{a}(t) = -\omega^2\vec{r}(t) \), of magnitude \( A\omega^2 \) and pointing opposite the position vector, toward the origin[6](https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-and-nonuniform-circular-motion).
If the speed also changes, as on a spinning top changing its spin rate or any accelerating rotor, a tangential acceleration \( a_T = \frac{d\lvert\vec{v}\rvert}{dt} \) appears, tangent to the circle. The centripetal and tangential components are perpendicular, and the total acceleration is their vector sum, pointing at an angle between them[6](https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-and-nonuniform-circular-motion). In a worked example with a particle on a circle of radius 2.0 m whose speed is 2.5 m/s at a given instant, the centripetal component is 3.1 m/s², the tangential component 1.5 m/s², and the total acceleration 3.47 m/s² at 64° from the tangent[6](https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-and-nonuniform-circular-motion).
## Relation to force
Kinematics describes acceleration without its causes; dynamics explains them. Under Newton's second law of motion, a system's acceleration points along the net external force applied to it and grows in proportion to that force, while varying inversely with the system's mass[7](https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law):
\[ \vec{a} = \frac{\vec{F}_{\text{net}}}{m}, \qquad \vec{F}_{\text{net}} = m\vec{a} \]
Experiments show the two proportionalities to be exact, and the acceleration of an object depends only on the net external force and its mass[7](https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law). In component form the law reads \( \sum F_x = ma_x \), \( \sum F_y = ma_y \), and \( \sum F_z = ma_z \)[7](https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law).
Newton originally stated the law in terms of momentum: the net force equals the instantaneous rate of change of a body's momentum, \( \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} \) with \( \vec{p} = m\vec{v} \). When the mass is constant this reduces to \( \vec{F}_{\text{net}} = m\vec{a} \); the momentum form remains valid when the mass changes, as in rocket motion[7](https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law).
## References
1. [3.3 Average and Instantaneous Acceleration, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration)
2. [NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI Prefixes](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes)
3. [Ch. 3 Key Equations, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/3-key-equations)
4. [NIST Guide to the SI, Appendix B.9: Factors for units listed by kind of quantity or field of science](https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b9)
5. [3.4 Motion with Constant Acceleration, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration)
6. [4.4 Uniform and Nonuniform Circular Motion, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-and-nonuniform-circular-motion)
7. [5.3 Newton's Second Law, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law)
8. [3.6 Finding Velocity and Displacement from Acceleration, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/3-6-finding-velocity-and-displacement-from-acceleration)
9. [4.3 Projectile Motion, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion)
10. [Motion in a Circle, Lesson Plan #29, NASA GSFC](https://cdaweb.gsfc.nasa.gov/pub/documents/archived_websites/pwg.gsfc.nasa.gov/stargaze/Lcircul.htm)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Motion, forces, and dynamics › Kinematics › Particle kinematics*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026; Oct 25, 2026 · Last review: Oct 15, 2026*
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