Question

The energy of a system as a function of time t is given as $E(t)={{A}^{2}}{{e}^{-\alpha t}}$, where $\alpha =0.2{{s}^{-1}}$. 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

Hint: Let us find the uncertainty in the energy equation with respect to time at any instant. As the percentage error to be found is at time t=5sec, substitute the value of time t=5sec in the uncertainty equation and solve for the error in the energy. The uncertainty in the value of A and t is given.

Complete step by step answer:
Let us write the given information, the energy of the system changes with respect to time t as,
$E(t)={{A}^{2}}{{e}^{-\alpha t}},\alpha =0.2{{s}^{-1}}$
The uncertainty in the A and the time t is 1.25, 1.5 respectively.
New, the uncertainty in the equation is given as,
$\begin{align}
  & \dfrac{dA}{dt}=1.25 \\
 & \dfrac{dt}{t}=1.5 \\
 & \\
\end{align}$
Now, the terms present in the power terms will be multiplied to the base when uncertainty is considered. So, the terms like alpha, time taken for the change in energy, and the constant present as power of A will be multiplied to respective bases.
$\begin{align}
  & \dfrac{dE}{dt}=\pm 2\dfrac{dA}{A}\pm \alpha \dfrac{dt}{t} \\
 & \Rightarrow \dfrac{dE}{dt}=\pm 2.5\pm 1.5 \\
 & \therefore \dfrac{dE}{dt}=\pm 4\% \\
\end{align}$
Therefore, in this way, the uncertainty in the energy is found.

Additional information:
Percent errors indicate how big our errors are when we measure something in an analysis process. Smaller percent errors indicate that we are close to the accepted or original value. For example, a 1% error indicates that we got very close to the accepted value, while 48% means that we were quite a long way off from the true value. Measurement errors are often unavoidable due to certain reasons like hands can shake, material can be imprecise, or our instruments just might not have the capability to estimate exactly. Simply, the percentage error is the difference between the estimate value and the actual value in comparison to the actual value and is expressed as a percentage.

Note:
In the above question, while calculating the percentage error or the uncertainty, the terms having a square will don’t get their uncertainty values squared, instead, the uncertainty gets multiplied by the value of the power of the quantity. This only goes for multiplication and division. In the case of addition and subtraction, the constants get zero after percentage error calculation.