PNC-01-080 – Instantaneous Acceleration

Instantaneous Acceleration

Instantaneous acceleration for one-dimensional motion:

The average acceleration vector describes how fast the velocity vector changes, on average, over a finite amount of time. However, it fails to adequately describe the acceleration of the particle at any given instant because the information between time $t_i$ and $t_f=t_i+\mathrm{\Delta }t$ is not recorded. To define the instantaneous acceleration, the time interval $\mathrm{\Delta }t$ between two consecutives measurements must shrink to an infinitely small amount as described below.

We define the instantaneous acceleration as the acceleration of an object at any instant $t$ and we denote it by $a_x\left(t\right)$. Mathematically, we achieve this by making the time interval $\mathrm{\Delta }t$ in the average acceleration formula infinitely small (shrinking it to zero without every reaching zero) which turns the average acceleration into the instantaneous acceleration.

Properties of instantaneous acceleration:

• A positive instantaneous acceleration at an instant $t$ denotes an increase in the instantaneous velocity at that instant.

• A negative instantaneous acceleration at an instant $t$ denotes a decrease in the instantaneous velocity at that instant.