# Acceleration is Second Derivative of Displacement with respect to Time

## Theorem

The **acceleration** of a body $M$ is the second derivative of the displacement of $M$ from a given point of reference with respect to time:

- $\mathbf a = \dfrac {\mathrm d^2 \mathbf s}{\mathrm d t^2}$

## Proof

By definition, the acceleration of a body $M$ is defined as the first derivative of the velocity of $M$ relative to a given point of reference with respect to time:

- $\mathbf a = \dfrac {\mathrm d \mathbf v}{\mathrm d t}$

Also by definition, the velocity of $M$ is defined as the first derivative of the displacement of $M$ from a given point of reference with respect to time:

- $\mathbf v = \dfrac {\mathrm d \mathbf s}{\mathrm d t}$

That is:

- $\mathbf a = \dfrac {\mathrm d}{\mathrm d t} \left({\dfrac {\mathrm d \mathbf s}{\mathrm d t}}\right)$

Hence the result by definition of the second derivative.

$\blacksquare$

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World