Instantaneous Acceleration

Instantaneous acceleration, \(a\) or the acceleration at a specific instant in time, is obtained by the same process as discussed for instantaneous velocity in the previous lesson on velocity and speed—that is, by considering an infinitesimally small interval of time.

How do we find instantaneous acceleration using only algebra? The answer is that we choose an average acceleration that is representative of the motion. The figure below shows graphs of instantaneous acceleration versus time for two very different motions.

In the (a) part of the figure, the acceleration varies slightly and the average over the entire interval is nearly the same as the instantaneous acceleration at any time. In this case, we should treat this motion as if it had a constant acceleration equal to the average (in this case about \(1.8\mathrm{\; m/s^2}\)).

In the (b) part of the figure, the acceleration varies drastically over time. In such situations it is best to consider smaller time intervals and choose an average acceleration for each. For example, we could consider motion over the time intervals from 0 to 1.0 s and from 1.0 to 3.0 s as separate motions with accelerations of \(+3.0\mathrm{\; m/s^2}\) and \(-2.0\mathrm{\; m/s^2}\), respectively.

Graphs of instantaneous acceleration versus time for two different one-dimensional motions. (a) Here acceleration varies only slightly and is always in the same direction, since it is positive. The average over the interval is nearly the same as the acceleration at any given time. (b) Here the acceleration varies greatly, perhaps representing a package on a post office conveyor belt that is accelerated forward and backward as it bumps along. It is necessary to consider small time intervals (such as from 0 to 1.0 s) with constant or nearly constant acceleration in such a situation. Image credit: OpenStax, College Physics