Terminal Velocity of a Re-entry Capsule (Aerospace Engineering) Solution

Part 1: Exponential Growth Dominance

As $t \to \infty$ , the exponential terms $e^{0.4t}$ grow exponentially without bound, rendering the additive constants $-12$ and $+5$ completely negligible.

Part 2: Analytical Limit Calculation

Divide both numerator and denominator by the dominant exponential term in the denominator, $e^{0.4t}$ :

\lim_{t \to \infty} \frac{76e^{0.4t} - 12}{2e^{0.4t} + 5} = \lim_{t \to \infty} \frac{\frac{76e^{0.4t}}{e^{0.4t}} - \frac{12}{e^{0.4t}}}{\frac{2e^{0.4t}}{e^{0.4t}} + \frac{5}{e^{0.4t}}} = \lim_{t \to \infty} \frac{76 - 12e^{-0.4t}}{2 + 5e^{-0.4t}}

Since $\lim_{t \to \infty} e^{-0.4t} = 0$ :

\frac{76 - 0}{2 + 0} = \frac{76}{2} = 38\text{ m/s}

Part 3: Horizontal Asymptote & Structural Conclusion

Horizontal Asymptote: $y = 38$
Aerospace Engineering Significance: The capsule's descent speed stabilizes at an absolute terminal velocity of $38\text{ m/s}$ ( $\approx 136.8\text{ km/h}$ ). The parachute deployment actuators and Kevlar tether lines must be rated to withstand the aerodynamic drag forces associated with this maximum velocity threshold.

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Terminal Velocity of a Re-entry Capsule (Aerospace Engineering) Solution