Acceleration is Second Derivative of Displacement with respect to Time

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

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): acceleration
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): acceleration
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $8$: The System of the World: Newton