The centimeter per second squared (symbolized cm/s ^{2} or cm/sec ^{2} ) is the unit of acceleration vector magnitude in the centimeter-gram-second ( cgs ) or small-unit metric system . This quantity can be defined in either of two senses: average or instantaneous.

For an object traveling in a straight line, the average acceleration magnitude is obtained by evaluating the object's instantaneous linear speed (in centimeters per second) at two different points *t* _{1} and *t* _{2} in time, and then dividing the distance by the span of time *t* _{2} - *t* _{1} (in seconds). Suppose the instantaneous speed at time *t* _{1} is equal to *s* _{1} , and the instantaneous speed at time *t* _{2} is equal to *s* _{2} . Then the average acceleration magnitude *a* _{avg} (in centimeters per second squared) during the time interval [ *t* _{1} , *t* _{2} ] is given by:

*a* _{avg} = ( *s* _{2} - *s* _{1} ) / ( *t* _{2} - *t* _{1} )

Instantaneous acceleration magnitude is more difficult to intuit, because it involves an expression of motion over an arbitrarily small interval of time. Let *p* represent a specific point in time. Suppose an object is in motion at about that time. The average acceleration magnitude can be determined over increasingly short time intervals centered at *p* , for example:

[ *p* -4, *p* +4]

[ *p* -3, *p* +3]

[ *p* -2, *p* +2]

[ *p* -1, *p* +1]

[ *p* -0.5, *p* +0.5]

[ *p* -0.25, *p* +0.25]

.

.

.

[ *p* - *x* , *p* + *x* ]

.

.

.

where the added and subtracted numbers represent seconds. The instantaneous acceleration magnitude, *a* _{inst} , is the limit of the average acceleration magnitude as *x* approaches zero. This is a theoretical value, because it can be obtained only by inference from instantaneous speed values determined at the starting and ending points of progressively shorter time spans.

Acceleration, in its fullest sense, is a vector quantity, possessing direction as well as magnitude. For an object moving in a straight line and whose linear speed changes, the acceleration vector points in the same direction as the object's direction of motion. But acceleration can be the result of a change in the direction of a moving object, even if the instantaneous speed remains constant. The classic example is given by an object in circular motion, such as a revolving weight attached to the rim of a wheel. If the rotational speed of the wheel is constant, the weight's acceleration vector points directly inward toward the center of the wheel.

Also see acceleration , velocity , International System of Units ( SI ), and Table of Physical Units and Constants.

*This was last updated in*June 2010

*Posted by:*Margaret Rouse

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