Question

A student measures that distance traversed in free fall of a body, initially at rest in a given time. He uses this data to estimated g, the acceleration due to gravity. If the maximum percentage error in measurement of the distance and the time are ${{e}_{1}}\text{ and }{{e}_{2}}$ respectively, the percentage error in the estimation of g is:
$\begin{aligned} & \text{A}\text{. }{{e}_{1}}\text{-}{{e}_{2}} \\\ & \text{B}\text{. }{{e}_{1}}+2{{e}_{2}} \\\ & \text{C}\text{. }{{e}_{1}}+{{e}_{2}} \\\ & \text{D}\text{. }{{e}_{1}}-2{{e}_{2}} \\\ \end{aligned}$

Answer 1

We will use the equation of motion for a free fall of body. From there we will obtain the equation of g. We will further solve it by taking log and differentiating. After that we will find the error percentage and will obtain the answer.

Formula used:
Formula of height for a free fall of body
$h=\dfrac{1}{2}g{{t}^{2}}$

Complete step by step answer:
We know that,
$\begin{aligned} & h=\dfrac{1}{2}g{{t}^{2}} \\\ & =g=\dfrac{2h}{{{t}^{2}}} \\\ \end{aligned}$
Taking log both the sides
$\log \left( g \right)=\log 2+\log h+2\log t$
Differentiating both the sides.
$\dfrac{\Delta g}{g}=0+\dfrac{\Delta h}{h}+2\dfrac{\Delta t}{t}$
Now to calculate percentage multiply both the sides by 100
$\dfrac{\Delta g}{g}\times 100=\left( 0+\dfrac{\Delta h}{h}+2\dfrac{\Delta t}{t} \right)\times 100$
Putting values we obtain
$={{e}_{1}}+2{{e}_{2}}$
So the percentage error in the estimation of g is ${{e}_{1}}+2{{e}_{2}}$.
Therefore the correct option is B.
Additional information:
There is error within the calculation and error within the measurement. Error within the calculations are either because the models are naive or because the measurements going into them were imprecise. For instance, a naive model maybe, didn't account for friction, or didn't account for the turbulence of the air because the object passes through it, etc. we can't actually account for each factor, though you'll generally get close enough.

Note:
Every measurement involve some error and this error are often expressed in three way,
Absolute error. Relative error and percentage error :
For percentage error we ought to have idea about above three terms, we'll see below relations or methods to find out percentage error,
Formula for Percentage (%) error = relative error ×100
Formula for Relative error = absolute error/ true value of observation.
Formula for absolute error = mean of individual deviation of observation from true value
These are the fundamental formulas to find errors.