Acceleration is the rate of change of velocity. At any point on a speed-time graph, the magnitude of the acceleration is given by the gradient of the tangent to the curve at that point.

In kinematics, acceleration is defined as the first derivative of velocity with respect to time (that is, the rate of change of velocity), or equivalently as the second derivative of position. It is a vector quantity with dimension L T^?2. In SI units, acceleration is measured in metres per second squared (m/s²).

In common speech, the term acceleration is only used for an increase in speed (the magnitude of velocity); a decrease in speed is called deceleration. In physics, any increase or decrease in speed is referred to as acceleration, and also a change in the direction of velocity is an acceleration (the centripetal acceleration; whereas the rate of change of speed is the tangential acceleration).

In classical mechanics, the acceleration of a body is proportional to the resultant (total) force acting on it (Newton's second law):

\mathbf{F} = m\mathbf{a} \quad \to \quad \mathbf{a} = \mathbf{F}/m

where F is the resultant force acting on the body, m is the mass of the body, and a is its acceleration.

Definition

Acceleration is a vector quantity which is defined as the rate at which an object changes its velocity.

An object is accelerating if it is changing its velocity. The formula for the average acceleration over a time interval Δt is

\mathbf{\bar{a}}=\frac{\mathbf{v}(t+\Delta t)-\mathbf{v}(t)}{\Delta t}

where v(t + Δt) is the final (instantaneous) velocity, v(t) is the initial velocity, t is the initial time and Δt is the duration of the interval.

The instantaneous acceleration at time t is defined as the limit of average acceleration as Δt approaches zero:

\mathbf{a}(t)=\lim_{\Delta t \to 0}\frac{\mathbf{v}(t+\Delta t)-\mathbf{v}(t)}{\Delta t}=\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}

which shows that a(t) is the first derivative of velocity. Since instantaneous velocity is in turn the first derivative of the position vector, acceleration is the second derivative of position:

\mathbf{a} = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}

where a is acceleration, v is velocity, r is position and t is time.

Relation to relativity

After completing his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant proper acceleration are indistinguishable from those in a gravitational field. This was the basis for his development of general relativity, a relativistic theory of gravity. This is also the basis for the popular twin paradox, which asks why one twin ages less when moving away from his sibling at near light-speed and then returning, since the non-aging twin can say that it is the other twin that was moving. General relativity solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). In special relativity, only inertial frames of reference (non-accelerated frames) can be used and are equivalent; general relativity considers all frames, even accelerated ones, to be equivalent. (The path from these considerations to the full theory of general relativity is traced in the introduction to general relativity.)

See also

• Uniform acceleration
• Angular acceleration
• Coordinate vs. physical acceleration
• Derivatives of position
• Equations of Motion
• Proper Acceleration
• 0 to 60 mph
• Shock (mechanics)

External links

• Acceleration and Free Fall - a chapter from an online textbook
• ''Trajectories and Radius, Velocity, Acceleration'' on Project PHYSNET (ERROR - PAGE MOVED)
• Science aid: Movement
• The physics classroom
• Science.dirbix: Acceleration
• Acceleration Calculator
• Motion Characteristics for Circular Motion
• Practical Guide to Accelerometers

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