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Acceleration (physics)

Change in velocity per unit time; a vector quantity describing how quickly speed or direction changes, with uses from free fall to vehicle design and measurement by accelerometers.

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Overview

Acceleration is the rate at which an object's velocity changes with respect to time. It describes how quickly speed increases or decreases and how direction alters. As a vector, acceleration conveys both magnitude and direction: the same change in speed can have different physical effects depending on which way it acts.

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Definitions and formulas

In everyday and scientific use, acceleration is often expressed as the change in velocity divided by the interval over which that change occurs. Average acceleration is given by the ratio of the change in velocity to the change in time, often written verbally as Δv / Δt. Instantaneous acceleration is the limit of this ratio as the time interval becomes infinitesimally small and corresponds to the time derivative of velocity in calculus.

Because acceleration is a vector, it can be described by components along coordinate axes or by a magnitude and direction. Common units are metres per second squared (m/s2) in the SI system. Related concepts include velocity (speed with direction) and time, both of which appear in the defining relationship. The rate at which acceleration itself changes is called jerk.

Types and examples

  • Uniform acceleration: when acceleration stays constant, motion equations simplify and are used for examples such as objects in free fall (neglecting air resistance) or a car accelerating at steady throttle.
  • Radial or centripetal acceleration: a change in direction while speed remains constant, as for an object moving in a circle; this acceleration points toward the center of curvature.
  • Non-uniform acceleration: acceleration that varies over time, producing changing forces and often noticeable as 'jerk' during rapid starts or stops.

Measurement and applications

Accelerometers detect acceleration and are used in navigation, vehicle safety systems, mobile devices, and vibration monitoring. Modern sensors, including MEMS accelerometers in phones, measure accelerations along one or more axes and combine data with other sensors to infer orientation and motion. Engineers use acceleration to design structures and vehicles to meet comfort and safety limits, while scientists use it to study dynamics in systems from biomechanics to astrophysics.

History and significance

The need to quantify acceleration emerged with the development of kinematics and calculus. Fundamental physical laws connect acceleration to the causes of motion: for example, classical mechanics relates force to acceleration, indicating that net forces produce proportional changes in velocity. The concept remains central across physics and engineering because it links how things move with the influences that produce that motion.

Distinctions and notable facts

Acceleration differs from speed and from simple displacement because it focuses on the change of velocity rather than on distance travelled. It is not enough to know how fast an object moves; knowing how its velocity changes is essential for predicting future motion. The mathematical and practical distinction between average and instantaneous acceleration is important in both theoretical problems and real-world measurements. For more discussion of related terms see displacement and time and the conceptual difference between a scalar and a vector.

Further reading and resources can clarify examples and computations: introductory texts cover constant-acceleration formulas, while advanced treatments address variable acceleration, non-inertial frames, and relativistic effects at high speeds. For foundational definitions and applied examples see educational materials and instrument manuals that describe how accelerometers operate and are calibrated.

Calculation

See also: Linear motion

The acceleration {\vec {a}} is the change in velocity per time interval. It is easiest to calculate when the acceleration is constant. If the velocities v(t_{1})at time t_{1}and v(t_{2})at time t_{2}known, the acceleration within the time interval Δ is \Delta t=t_{2}-t_{1}calculated from the difference of the velocities Δ accordance with \Delta v=v(t_{2})-v(t_{1})

a={\frac {\Delta v}{\Delta t}}.

For a constant acceleration that does not {\vec {v}}occur in the direction of the velocity vector , the difference in velocities Δ must be \Delta {\vec {v}}={\vec {v}}(t_{2})-{\vec {v}}(t_{1})determined vectorially, as illustrated in the figure. If the acceleration changes during the time period under consideration, the above calculation yields the mean acceleration, also called the average acceleration.

In order to calculate the acceleration for a specific point in time instead of for a time interval, one must go from the difference quotient to the differential quotient. The acceleration is then the first time derivative of the velocity with respect to time:

{\vec {a}}(t)={\frac {\mathrm {d} {\vec {v}}(t)}{\mathrm {d} t}}={\dot {\vec {v}}}(t)

Since velocity is the derivative of location with respect to time, acceleration can also be {\vec {r}}represented as the second derivative of the location vector

{\vec {a}}(t)={\frac {\mathrm {d} ^{2}{\vec {r}}(t)}{\mathrm {d} t^{2}}}={\ddot {\vec {r}}}(t)

The time derivative of the acceleration (i.e., the third derivative of the location vector with respect to time) is called jerk : {\vec {j}}

{\vec {j}}(t)={\dot {\vec {a}}}(t)={\frac {\mathrm {d} ^{3}{\vec {r}}(t)}{\mathrm {d} t^{3}}}

Examples of calculation via speed

A car is moving at time t_{1}=0\,\mathrm {s} with a speed of v_{1}=10\,\mathrm {\tfrac {m}{s}} across the road (which is 36 km/h). Ten seconds later, at time t_{2}=10\,\mathrm {s} , the speed is v_{2}=30\,\mathrm {\tfrac {m}{s}} (which is 108 km/h). The average acceleration of the car in this time interval was then

{\displaystyle a={\frac {v_{2}-v_{1}}{t_{2}-t_{1}}}=2\,\mathrm {\frac {m}{s^{2}}} }.

The speed has thus increased by an average of 2 m/s (i.e. by 7.2 km/h) per second.

A passenger car driving before the red light within Δ {\displaystyle \Delta t=3\,\mathrm {s} }of "speed 50" ( {\displaystyle v_{1}=50\,\mathrm {\tfrac {km}{h}} \approx 14\,\mathrm {\tfrac {m}{s}} }) is decelerated to zero, the acceleration undergoes

{\displaystyle a={\frac {0-v_{1}}{\Delta t}}\approx -5\,\mathrm {\frac {m}{s^{2}}} }.

Unit of acceleration

The standard unit of measurement for specifying an acceleration is the unit meter per square second (m/s2), i.e. (m/s)/s. In general, loads on technical equipment or the specification of load limits can be expressed as g-force, i.e. as "force per mass". This is given as a multiple of the normal acceleration due to gravity (standard acceleration due to gravity) g = 9.80665 m/s2. In the geosciences, the unit Gal = 0.01 m/s2 is also commonly used.

Acceleration of motor vehicles

For motor vehicles, the achievable positive acceleration is used as an essential parameter for classifying the performance. An average value is usually given in the form "In ... seconds from 0 to 100 km/h" (also 60, 160 or 200 km/h).

Numerical example:

For the Tesla Model S (type: Performance), it is stated that acceleration from 0 to 100 km/h can be achieved in 2.5 seconds. This corresponds to an average acceleration value of

{\displaystyle a={\frac {\Delta v}{\Delta t}}={\frac {27{,}8\,\mathrm {\frac {m}{s}} }{2{,}5\,\mathrm {s} }}=11\,\mathrm {\frac {m}{s^{2}}} }.

Geometric construction of the difference of velocity vectors
Geometric construction of the difference of velocity vectors

Acceleration measurement

In principle, there are two ways of measuring or specifying accelerations. The acceleration of an object can be considered kinematically with respect to a path (space curve). For this purpose, the instantaneous velocity is determined, and its rate of change is the acceleration. The other possibility is to use an accelerometer. This determines the inertial force with the aid of a test mass, from which the acceleration is then inferred with the aid of Newton's basic equation of mechanics.

Questions and answers

Q: What is acceleration?

A: Acceleration is a measure of how fast velocity changes.

Q: How is acceleration measured?

A: Acceleration is the change of velocity divided by the change of time.

Q: What type of quantity is acceleration?

A: Acceleration is a vector, and therefore includes both a size and a direction.

Q: How is speed defined?

A: Speed is how fast you are moving, and is measured as distance traveled divided by time taken.

Q: What is the difference between speed and velocity?

A: Velocity is a vector quantity and refers to how quickly your position is changing and in what direction.

Q: What is displacement?

A: Displacement is how much your position has changed in what direction.

Q: What is jerk?

A: Jerk is the measurement of how fast acceleration changes.

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