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## Acceleration

The conventional definition of acceleration is as follows:
Acceleration is the rate of change of velocity with time.
This definition implies that (15)

where is the body's acceleration at time , and is the change in velocity of the body between times and .

How should we choose the time interval appearing in Eq. (15)? Again, in the simple case in which the body is moving with constant acceleration, we can make as large or small as we like, and it will not affect the value of . Suppose, however, that is constantly changing in time, as is generally the case. In this situation, must be kept sufficiently small that the body's acceleration does not change appreciably between times and .

A general expression for instantaneous acceleration, which is valid irrespective of how rapidly or slowly the body's acceleration changes in time, can be obtained by taking the limit of Eq. (15) as approaches zero: (16)

The above definition is particularly useful if we can represent as an analytic function, because it allows us to immediately evaluate the instantaneous acceleration via the rules of calculus. Thus, if is given by formula (11) then (17)

Figure 5 shows the graph of versus time obtained from the above expression. Note that when is positive the body is accelerating to the right (i.e., is increasing in time). Likewise, when is negative the body is decelerating (i.e., is decreasing in time). Fortunately, it is generally not necessary to evaluate the rate of change of acceleration with time, since this quantity does not appear in Newton's laws of motion.   Next: Motion with constant velocity Up: Motion in 1 dimension Previous: Velocity
Richard Fitzpatrick 2006-02-02