Question

The energy of a system as a function of time t is given as $E\left(t\right)=A^2{e}^{-\alpha t}$, where $\alpha =0.2s^{-1}$. The measurement of A has an error of 1.25%. If the error in the measurement of time is 1.50%, the find percentage error in the value of E(t) at $t=5$ s ?

Answer 1

$E\left(t\right)=A^2{e}^{-\alpha t}$
and $\alpha =0.2/s$

We have, $\frac{dA}{A}=1.25$

$\frac{dt}{t}=1.5$

Now, $logE=2logA-\alpha t$

Or, $\frac{dE}{E}=\pm 2\frac{dA}{A}\pm \alpha dt$

$=\pm 2\left(1.25\right)\pm 0.2\left(7.5\right)$

$=\pm 2\left(1.25\right)\pm 1.5$

$=\pm 4\%$