# The distance d that an object falls varies directly as the square of the time t during which it...

## Question:

The distance d that an object falls varies directly as the square of the time t during which it is falling. If an object falls 400 feet in 5 seconds, how long will it take the object to fall 9 feet?

## Direct Proportion:

If a quantity varies directly with another quantity then two can be said to be directly proportional to each other. Removing the proportionality between the two quantities we introduce the proportionality constant that links the two. Knowing one scenario where the numerical values of the two quantities are known, the proportionality constant can then be calculated, resulting in the determination of the full equation or relationship between the two.

{eq}\begin{align*} d &\propto t^2 & \text{[d is the distance, t is the time]}\\ \therefore d &= kt^2 & \text{[k is the proportionality constant]}\\ \Rightarrow k &= \frac{d}{t^2}\\ &= \frac{400}{25^2}\\ &= 16\ ft/s^2\\\\ \therefore d &= 16t^2 & \text{[leaving out the units for clarity]}\\ \end{align*} {/eq}

So, given d is 9 ft, the duration is calculated by using the above relationship, as follows;

{eq}\begin{align*} d &= 16t^2 & \text{[leaving out the units for clarity]}\\ \therefore t &= \frac{\sqrt{d}}{4}\\ &= \frac{\sqrt{9}}{4}\\ &= 0.75\ s \end{align*} {/eq}

So it would take 0.8 s