Acceleration

The conventional definition of acceleration is as follows:

Acceleration is the rate of change of velocity with time.

This definition implies that

$$a = \frac{{\mit\Delta}v}{{\mit\Delta} t},$$ (15)

where $a$ is the body's acceleration at time $t$, and ${\mit\Delta} v$ is the change in velocity of the body between times $t$ and $t+{\mit\Delta} t$.

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

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 ${\mit\Delta} t$ approaches zero:

$$a = \lim_{{\mit\Delta}t\rightarrow 0}\frac{{\mit\Delta}v}{{\mit\Delta} t}= \frac{dv}{dt}=\frac{d^2 x}{dt^2}.$$ (16)

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

$$a = \frac{d^2 x}{dt^2} = 1 - 3 t^2.$$ (17)

Figure 5 shows the graph of $a$ versus time obtained from the above expression. Note that when $a$ is positive the body is accelerating to the right (i.e., $v$ is increasing in time). Likewise, when $a$ is negative the body is decelerating (i.e., $v$ is decreasing in time).

Figure 5: Graph of instantaneous acceleration versus time associated with the motion specified in Fig. 3

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.