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Rocket Payload Velocity (Mechanical & Aerospace Engineering) Solution

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Part 1: Limit Law Applied

Since both lim⁡t→4T(t)\lim_{t\to 4} T(t) and lim⁡t→4M(t)\lim_{t\to 4} M(t) exist as finite values, we apply the Product Law for Limits, which states that the limit of a product is the product of the limits:

lim⁡t→4V(t)=lim⁡t→4[T(t)⋅M(t)]=(lim⁡t→4T(t))⋅(lim⁡t→4M(t))\lim_{t\to 4} V(t) = \lim_{t\to 4} [T(t) \cdot M(t)] = \left(\lim_{t\to 4} T(t)\right) \cdot \left(\lim_{t\to 4} M(t)\right)

Part 2: Calculation

Substitute the given telemetry limits:

lim⁡t→4V(t)=(8 m/s2)⋅(12)=96 m/s\lim_{t\to 4} V(t) = (8\text{ m/s}^2) \cdot (12) = 96\text{ m/s}

Conclusion: The expected payload velocity as the rocket approaches the 4-second stage transition is 96 m/s96\text{ m/s}. The Product Law allows aerospace engineers to decouple complex multi-variable propulsion dynamics into distinct, independently tested subsystem limits.