Average and Instantaneous Acceleration

Suppose that $\overrightarrow{x}(t)$ is the position function for a moving particle. In the previous section, we defined the average velocity from time $t=t_i$ to $t=t_f$ to be $$\overrightarrow{v}_{\text{av}}=\frac{\overrightarrow{x}(t_f)-\overrightarrow{x}(t_i)}{t_f-t_i},$$ and the instantaneous velocity at time $t=t_i$ to be $$\overrightarrow{v}_{\text{inst}}=\frac{d\overrightarrow{x}(t)}{dt}\bigg|_{t=t_i}.$$ We can create analogous definitions for acceleration, which measures how fast the velocity changes over time. For example, if $\overrightarrow{v}(t)$ is the velocity function, we define the average acceleration from $t=t_i$ to $t=t_f$ to be $$\overrightarrow{a}_{\text{av}}=\frac{\overrightarrow{v}(t_f)-\overrightarrow{v}(t_i)}{t_f-t_i}.$$

By the definition of average acceleration, $$\overrightarrow{a}_{\text{av}}=\frac{\overrightarrow{v}(2)-\overrightarrow{v}(1)}{2-1}=\frac{(3(2)^2+1)-(3(1)^2+1)}{2-1}=9.$$

Similar to instantaneous velocity, we can define instantaneous acceleration at $t=t_i$ as $$\overrightarrow{a}_{\text{inst}}=\frac{d\overrightarrow{v}(t)}{dt}\bigg|_{t=t_i}.$$

Let's look at another example:

Now you might think, since we extended velocity to acceleration, can we extend acceleration to something else the same way? The answer is yes, and this new quantity is called jerk. However, we will usually not go this far in calculus.