Question

A ball is dropped from a height. If the maximum percentage errors in measurement of the height and the time are $e_1$ and $e_2$ respectively, then maximum percentage error in the estimation of $g$ is

• A. $e_1-2e_2$
• B. $e_1+2e_2$
• C. $e_1+e_2$
• D. $e_2-e_1$

Answer 1

The correct option is B $e_1+2e_2$

From dimensional formula for acceleration due to gravity
$\left[g\right]=\left[L{T}^{-2}\right]$
$\Rightarrow \frac{\Delta g}{g}=\frac{\Delta L}{L}+\frac{2\Delta T}{T}$ (For maximum error in $g$)
$\Rightarrow \frac{\Delta g}{g}\times 100=[(\frac{\Delta L}{L}\times 100)+(\frac{2\Delta T}{T}\times 100)]$
Given, $\frac{\Delta L}{L}\times 100=e_1$ (error in height) and $\frac{\Delta T}{T}\times 100=e_2$ (error in time)
$\therefore \frac{\Delta g}{g}\times 100=e_1+2e_2$
Hence, option $\left(b\right)$ is the correct answer.