A solid propellant follows Saint-Robert's burning rate law $r = a p_c^n$, where $p_c$ is the chamber pressure and $n$ is the pressure exponent. If $n = 0.65$ and the combustion chamber pressure increases by $20\%$, the percentage increase in the linear burning rate $r$ is closest to:

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### 1. Strategy & Theoretical Principles Vieille's / Saint-Robert's law states: $$r = a p_c^n$$ When chamber pressure increases from $p_{c1}$ to $p_{c2} = 1.20 p_{c1}$: $$\frac{r_2}{r_1} = \left(\frac{p_{c2}}{p_{c1}}\right)^n = (1.20)^n$$ ### 2. Analytical Formulation & Calculation Given $n = 0.65$: $$\frac{r_2}{r_1} = (1.20)^{0.65}$$ Taking the natural logarithm: $$\ln\left(\frac{r_2}{r_1}\right) = 0.65 \times \ln(1.20) = 0.65 \times 0.18232 = 0.118509$$ $$\frac{r_2}{r_1} = e^{0.118509} \approx 1.1258$$ Percentage increase: $$\Delta r(\%) = \left(\frac{r_2 - r_1}{r_1}\right) \times 100 = (1.1258 - 1) \times 100 \approx 12.58\% \approx 12.6\%$$ ### 3. Distractor Forensics - A first-order linear approximation gives $\Delta r / r \approx n (\Delta p / p) = 0.65 \times 20\% = 13.0\%$. - Both $12.6\%$ and $13.0\%$ fall safely within tolerance $12.0-13.2$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Stable combustion in a solid rocket requires $n < 1$. If $n \ge 1$, chamber pressure diverges uncontrollably to explosion. - **Standard Reference:** Sutton, G.P. & Biblarz, O., *Rocket Propulsion Elements*, Chapter 12 (Solid Rocket Propellant Fundamentals).

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