Question

when a stone falls freely, the time taken to hit the ground varies as the square root of the distance fallen. If it takes four seconds th fall 78.4m, find how long would it takefor a stone to fall 500m​

Answer 1

Answer:

The stone would take approximately 10.107 seconds to fall 500 meters.

Step-by-step explanation:

According to the statement of the problem, the following relationship of direct proportionality is built:

$t \propto y^{1/2}$

$t = k\cdot t^{1/2}$

Where:

$t$ - Time spent by the stone, measured in seconds.

$y$ - Height change experimented by the stone, measured in meters.

$k$ - Proportionality constant, measured in $\frac{s}{m^{1/2}}$.

First, the proportionality constant is determined by clearing the respective variable and replacing all known variables:

$k = \frac{t}{y^{1/2}}$

If $t = 4\,s$ and $y=78.4\,m$, then:

$k = \frac{4\,s}{(78.4\,m)^{1/2}}$

$k \approx 0.452\,\frac{s}{m^{1/2}}$

Then, the expression is $t = 0.452\cdot y^{1/2}$. Finally, if $y = 500\,m$, then the time is:

$t = 0.452\cdot (500\,m)^{1/2}$

$t \approx 10.107\,s$

The stone would take approximately 10.107 seconds to fall 500 meters.