Question

You can calculate the time it will take for an object to fall from the top of a building to the ground if you know the height of the building. If the height has an uncertainty of 1.0% ,what is the uncertainty in time?

Short answer

The uncertainty in time is 0.5 %

Step-by-step solution

About Uncertainty for powers and roots

For the function $\mathrm{y}={\mathrm{x}}^{\mathrm{a}}$ the relative uncertainty in $\mathrm{y}(\%{\mathrm{e}}_{\mathrm{y}})$is t times the relative uncertainty in $\mathrm{x}(\%{\mathrm{e}}_{\mathrm{x}})$

$\mathrm{y}={\mathrm{x}}^{\mathrm{a}}$

$\Rightarrow \%{\mathrm{e}}_{\mathrm{y}}=\mathrm{a}\left(\%{\mathrm{e}}_{\mathrm{x}}\right)$

Calculate the amount of time it takes for an object to fall from a building to the ground

Given information:

An object falls from the top of a building to a known height on the ground.

The uncertainty in height is 1.0 %

Calculation of time uncertainty:

Consider an object falling from a height of$1/2{\mathrm{gt}}^2$in time t.

Now, use the required formula:

$$\begin{gathered} y=x^a \\ \Rightarrow \%e_y=a\left(\%e_x\right) \end{gathered}$$

Height $=1/2{\mathrm{gt}}^2$

Here, $a=$2

$$\begin{gathered} \%{\mathrm{e}}_{\mathrm{y}}=\mathrm{a}\left(\%{\mathrm{e}}_{\mathrm{x}}\right) \\ \%{\mathrm{e}}_{\mathrm{height}}=2(\%{\mathrm{e}}_{\mathrm{time}}) \\ 1.0\%=2\left(\%{\mathrm{e}}_{\mathrm{time}}\right) \end{gathered}$$

Now, rearrange the above formed value as shown,

$$\begin{gathered} \left(\%{\mathrm{e}}_{\mathrm{time}}\right)=\frac{1.0\%}{2} \\ =0.5\% \end{gathered}$$

Therefore, the uncertainty in time is 0.5%