# Application of the Second Derivative (Acceleration)

## Key Questions

• Let us first find the velocity function $v \left(t\right)$.
$v \left(t\right) = s ' \left(t\right) = 3 {t}^{2} + 6 t$

Let us now find the acceleration function $a \left(t\right)$.
$a \left(t\right) = v ' \left(t\right) = s ' ' \left(t\right) = 6 t + 6$

Acceleration is change in velocity over time $A = \frac{V}{T}$

#### Explanation:

For example lets say you have a car and its 0-60 MPH time is 3 seconds so the equation is $A = \frac{60}{3}$ so you would take 60 and divide it by three and that is 20 so the 0-60 speed is 20 seconds.

If you have a position function $x \left(t\right)$, then the derivative is a velocity function $v \left(t\right) = x ' \left(t\right)$ and the second derivative is an acceleration function $a \left(t\right) = x ' ' \left(t\right)$.