# Acceleration is Second Derivative of Displacement with respect to Time

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## Theorem

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

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

## Proof

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

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

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:

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

That is:

- $\mathbf a = \map {\dfrac \d {\d t} } {\dfrac {\d \mathbf s} {\d t} }$

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: Newton