Distance Moved by Body from Rest under Constant Acceleration

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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 theconstant 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.


Sources