# Acceleration Measure

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An Acceleration Measure is a physical measure of a rate of change of velocity with respect to time of a physical object .

**AKA:**Instantaneous Acceleration.**Context:**- It is a vector quantity.
- It can be defined as the derivative of velocity ([math]\displaystyle{ \vec{v} }[/math]) and the second derivative of distance([math]\displaystyle{ \vec{s} }[/math]) with respect to time

- [math]\displaystyle{ \vec{a}=d\vec{v}/dt=d^2\vec{s}/dt^2 }[/math].
- The units of measurement of acceleration are given by:

- [math]\displaystyle{ [acceleration]=\frac{[velocity]}{[time]}=\frac{[distance]}{[time^2]}=\frac{[length]}{[time^2]} }[/math]
- where [math]\displaystyle{ [x] }[/math] symbolizes the conversion of the quantity [math]\displaystyle{ x }[/math] to its units of measurement. Thus, the units of measurement for acceleration can be given in meters per squared seconds, [math]\displaystyle{ m/s^2 }[/math], (in International System of Units or metric system) or feet per squared seconds (imperial system).

- On average the acceleration can be estimated to be:

- [math]\displaystyle{ \vec{a}_{average}=\vec{\bar{a}}=\frac{\Delta v}{\Delta t}=\frac{\vec{v_2}-\vec{v_1}}{t_2-t_1} }[/math]
- Newton's Second Law shows that there is direct proportionality of acceleration to force and the inverse proportionality of acceleration to mass

- [math]\displaystyle{ \vec{F} =m\vec{a} \quad\iff\quad \vec{a}=\frac{\vec{F}}{m} }[/math]
- (Note: this is assuming object's mass is a constant, relativistic effects have to be taken into account when the object approaches the speed of light).

**Example(s)****Counter-Example(s)****See:**Motion, Momentum, Velocity, Force, Newton's Second Law, Free-fall Acceleration, Angular Acceleration, Tangential Acceleration, Centripetal Acceleration.

## References

### 2016

- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/acceleration Retrieved:2016-4-29.
**Acceleration**, in physics, is the rate of change of velocity of an object. An object's acceleration is the net result of any and all forces acting on the object, as described by Newton's Second Law. The SI unit for acceleration is metre per second squared (m s^{-2}). Accelerations are vector quantities (they have magnitude and direction) and add according to the parallelogram law. As a vector, the calculated net force is equal to the product of the object's mass (a scalar quantity) and its acceleration. For example, when a car starts from a standstill (zero relative velocity) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the car turns, there is an acceleration toward the new direction. In this example, we can call the forward acceleration of the car a "linear acceleration", which passengers in the car might experience as a force pushing them back into their seats. When changing direction, we might call this "non-linear acceleration", which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the opposite direction from the direction of the vehicle, sometimes called**deceleration**. Passengers may experience deceleration as a force lifting them forwards. Mathematically, there is no separate formula for deceleration: both are changes in velocity. Each of these accelerations (linear, non-linear, deceleration) might be felt by passengers until their velocity (speed and direction) matches that of the car.

### 2005

- (Hyperphysics Encyclopedia, 2005) ⇒ http://hyperphysics.phy-astr.gsu.edu/hbase/acca.html#c1
- QUOTE: Acceleration is defined as the rate of change of velocity. Acceleration is inherently a vector quantity, and an object will have non-zero acceleration if its speed and/or direction is changing. The average acceleration is given by

- [math]\displaystyle{ \vec{a}_{average}=\vec{\bar{a}}=\frac{\Delta v}{\Delta t}=\frac{\vec{v_2}-\vec{v_1}}{t_2-t_1} }[/math]
- where the small arrows indicate the vector quantities. The operation of subtracting the initial from the final velocity must be done by vector addition since they are inherently vectors.
- The units for acceleration can be implied from the definition to be meters/second divided by seconds, usually written m/s^2.
- The instantaneous acceleration at any time may be obtained by taking the limit of the average acceleration as the time interval approaches zero. This is the derivative of the velocity with respect to time:

- [math]\displaystyle{ \vec{a}_{instantaneous}=\lim_{\Delta\;t\rightarrow\;0}\frac{\Delta \vec{v}}{\Delta t}=\frac{d\vec{v}}{dt} }[/math]

### 1963

- (Feynman et al., 1963) ⇒ Richard P. Feynman, Robert B. Leighton and Matthew Sands (1963, 1977, 2006, 2010, 2013) "The Feynman Lectures on Physics": New Millennium Edition is now available online by the California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer ⇒ http://www.feynmanlectures.caltech.edu/
- QUOTE: The next step in developing the equations of motion is to introduce another idea which goes beyond the concept of velocity to that of change of velocity, and we now ask, “How does the velocity change?”. In previous chapters we have discussed cases in which forces produce changes in velocity. (...) In other words, by how many feet per second does the velocity change in a second, that is, how many feet per second, per second?(...) we want to find out how much the velocity changes per second; this quantity is called the acceleration.

- Acceleration is defined as the time rate of change of velocity. From the preceding discussion we know enough already to write the acceleration as the derivative [math]\displaystyle{ dv/dt }[/math], in the same way that the velocity is the derivative of the distance(...)We have another law that the velocity is equal to the integral of the acceleration. This is just the opposite of [math]\displaystyle{ a=dv/dt }[/math]; we have already seen that distance is the integral of the velocity, so distance can be found by twice integrating the acceleration.