Rocket Drag and Aerodynamic Deceleration (Aerospace Engineering) Solution

Problem Overview

We analyze the drag sensitivity gradient $\frac{dF_D}{dv}$ for the aerospace model:

F_D(v) = 0.002 v^2 + 180 v^{-\frac{1}{3}} + 0.05 v^{\frac{3}{2}}

Step-by-Step Mathematical Solution

Step 1: Pre-Process into Explicit Power Form

Convert the cube root denominator using negative rational exponents:

\frac{180}{\sqrt[3]{v}} = 180 v^{-\frac{1}{3}}

F_D(v) = 0.002 v^2 + 180 v^{-\frac{1}{3}} + 0.05 v^{\frac{3}{2}}

Step 2: Differentiate Term-by-Term via Power Rule

\frac{dF_D}{dv} = 0.002(2v) + 180\left(-\frac{1}{3}v^{-\frac{4}{3}}\right) + 0.05\left(\frac{3}{2}v^{\frac{1}{2}}\right)

\frac{dF_D}{dv} = 0.004 v - 60 v^{-\frac{4}{3}} + 0.075 v^{\frac{1}{2}}

Step 3: Evaluate at $v = 64\text{ m/s}$

Substitute $v = 64 = 4^3 = 8^2$ :

$0.004(64) = 0.256$
$60 (64)^{-\frac{4}{3}} = 60 (4^{-4}) = \frac{60}{256} \approx 0.234375$
$0.075 (64)^{\frac{1}{2}} = 0.075(8) = 0.600$

\left.\frac{dF_D}{dv}\right|_{v=64} = 0.256 - 0.234375 + 0.600 = 0.621625\text{ kN/}(\text{m/s})

Engineering Conclusion

The drag sensitivity gradient $\frac{dF_D}{dv}$ indicates that around $64\text{ m/s}$ , every additional meter per second increase in flight speed adds approximately $0.622\text{ kN}$ of drag resistance. This derivative directly feeds the rocket's adaptive throttle control algorithm to prevent structural airframe buffeting.

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