2 MOTION IN 1 DIMENSION 2.4 Acceleration
Figure 5: Graph of instantaneous acceleration versus time associated with the motion specified in
Fig. 3
by taking the limit of Eq. (2.5) as ∆t approaches zero:
a = lim
∆v dv
= = d(^2) x
∆t (^0) ∆t dt dt^2.^ (2.6)^
The above definition is particularl
→
y useful if we can represent x(t) as an analytic
function, because it allows us to immediately evaluate the instantaneous acceler-
ation a(t) via the rules of calculus. Thus, if x(t) is given by formula (2.1) then
d^2 x
a =
dt^2
= 1 − 3t^2
. (2.7)
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 in-
creasing in time). Likewise, when a is negative the body is decelerating (i.e., v is
decreasing in time).
Fortunately, it is generally not necessary to evaluate the rate of change of ac-celeration with time, since this quantity does not appear in Newton’s laws of
motion.