# Distance Moved by Body from Rest under Constant Acceleration

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## Contents

## Theorem

Let a body $B$ be stationary.

Let $B$ be subject to a constant acceleration.

Then the distance travelled by $B$ is proportional to the square of the length of time $B$ is under the acceleration.

## Proof

From Body under Constant Acceleration: Distance after Time:

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

where:

- $\mathbf s$ is the displacement of $B$ at time $t$ from its initial position at time $t$
- $\mathbf u$ is the velocity at time $t = 0$
- $\mathbf a$ is the constant acceleration $t$

In this scenario, $\mathbf u = \mathbf 0$.

Thus:

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

and so by taking the magnitudes of the vector quantities:

- $s = \dfrac a 2 t^2$

Hence the result, by definition of proportional.

$\blacksquare$

## Historical Note

This result was noted by Galileo Galilei by observation and reasoning.

It was confirmed mathematically by Isaac Newton by application of his laws of motion.

He published it in his *Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze*.

## Sources

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