Acceleration

Definition: Average Acceleration

The average acceleration of a point mass during a time interval $\Delta t$ is the ratio of the change in its velocity between the moments $t_1$ and $t_2>t_1$ and the interval’s duration:

$${\mathbf{a}}_{\text{avg}}\stackrel{\text{def}}{=}\frac{\mathbf{v}(t_2)-\mathbf{v}(t_1)}{t_2-t_1}=\frac{\Delta \mathbf{v}}{\Delta t}$$

Definition: Instantaneous Acceleration

The instantaneous acceleration of point mass at a moment $t_0$ is its average velocity within an infinitely small time interval after $t_0$:

$$\mathbf{a}(t_0)\stackrel{\text{def}}{=}\underset{\Delta t\to 0}{\lim }{\mathbf{a}}_{\text{avg}}=\underset{\Delta t\to 0}{\lim }\frac{\mathbf{v}(t_0+\Delta t)-\mathbf{v}(t_0)}{\Delta t}=\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}(t_0)$$

NOTE

The instantaneous acceleration is just the temporal derivative of velocity.

NOTE

When one says “acceleration”, they usually mean the instantaneous acceleration.

\