## 2 Materials and Methods
### Calculation of Chain Measurement Uncertainty  
The instruments and equipment used in this process have inherent uncertainties. Therefore, the conversion factors must reflect these specific uncertainties in the measurement chain to ensure the process's metrological reliability. This investigation is important \begin{table}
\begin{tabular}{c c} \hline
**Coefficient** & **Value** \\ \hline \(C_{f}\) & 0.004 mV/N \\ \(V_{0}\) & 0.1306 mV \\ \hline \end{tabular}
\end{table} 
Table 1: Coefficients of the mathematical model. because conversion coefficient errors directly impact the thrust results obtained after firing tests. This study explores the calculation of uncertainty involved in converting raw thrust data (voltage in \(V\)) into engineering units (force in \(N\)). It also addresses the numerical integration of the thrust curve to determine the total impulse, as described in the Subsection \"Measurement Chain\". To assess the uncertainty associated with the conversion factor, concepts of combined uncertainty and expanded uncertainty will be applied. According to the JCGM (2008) [8], uncertainties of input quantities are classified as Type A and Type B. Type A uncertainty is assessed through the statistical analysis of a series of repeated measurements during the test. In contrast, Type B uncertainty is a measure of measurement uncertainty based on information sources that do not rely on repeated observations, as is the case with Type A uncertainty. It is estimated from data such as instrument manufacturer specifications, calibration certificates, technical literature, or expert judgment. This uncertainty arises from various sources, such as instrument precision, calibration errors, and environmental factors that can affect the measurement process. Regarding Type A uncertainty, by analyzing the spread of data collected during multiple trials, the experimental standard deviation provides a quantitative measure of how much individual measurements deviate from the mean value, thereby offering insight into the reliability and accuracy of the measurement system. In this context, the experimental standard deviation of the measurement chain serves as a measure of uncertainty associated with the mean value, indicating the repeatability of the measurement, as shown in Equation (6), where \(S_{\text{\tiny{T}}}\) is the standard deviation of the mean, \(S\) is the standard deviation, and n is the number of samples. \[S_{\text{\tiny{T}}}=\frac{S}{\sqrt{n}} \tag{6}\]  
On the other hand, Type B uncertainty assessment uses approaches distinct from statistical analyses of a series of repeated measurements conducted during the test. According to the GUM (JCGM 2008) [8], this Type B evaluation relies on other types of knowledge, such as historical data on measurement method performance, uncertainties associated with equipment and standards calibration, technical specifications provided by manufacturers, and prevailing environmental conditions. Consider quantity y that is dependent on other N quantities through function f, as follows:  
\[y=f(x_{1},x_{2},x_{3},\dots,x_{N}) \tag{7}\]  
When the input quantities are not correlated, combined uncertainty is calculated as expressed in Equation (8):  
\[{u_{c}}^{2}(y)=\sum\biggl{[}\frac{\partial f}{\partial x_{i}}u(x_{i})\biggr{]} ^{2} \tag{8}\]  
where \(u_{c}\) is the combined uncertainty, \(\frac{\partial f}{\partial x_{i}}\) are the sensitivity coefficients, and \(u(x_{i})\) is the standard uncertainty of each input variable. To calculate the effective degrees of freedom (\(v_{eff}\)), ISOSUM [8] suggests using Equation (9), also known as the Welch-Satterthwaite formula. In this equation, \(v_{i}\) represents the degrees of freedom corresponding to each source of uncertainty. \[v_{eff}=\frac{u_{c}^{4}(y)}{\sum\frac{u_{i}^{4}(y)}{v_{i}}} \tag{9}\]  
The expanded uncertainty provides an interval around the measurement result within which the true value is expected to lie with a specified level of confidence.