p(t) graph of a rocket

Problem

Graph reading: $p(t)$ of a rocket. A rocket fires its engines at $t = 0$. The graph of $p(t)$ is a rising straight line for $30\;\text{s}$, then flat for $20\;\text{s}$, then decreasing for $10\;\text{s}$. Interpret the three regimes in light of $dp/dt = F_\text{net}$.

Solution

The key is Newton’s second law in momentum form: the slope of the $p(t)$ graph is the net force. $\frac{dp}{dt} = F_\text{net}$

Phase 1 ($0 \to 30\;\text{s}$), rising straight line: positive, constant slope. $\frac{dp}{dt} > 0 \;\text{(constant)} \;\Rightarrow\; F_\text{net} > 0 \;\text{constant}$ The engines provide constant thrust: momentum increases linearly.

Phase 2 ($30 \to 50\;\text{s}$), flat section: zero slope. $\frac{dp}{dt} = 0 \;\Rightarrow\; F_\text{net} = 0$ Engines off and external forces negligible: the system is isolated and $p$ is conserved (stays constant).

Phase 3 ($50 \to 60\;\text{s}$), decreasing straight line: negative slope. $\frac{dp}{dt} < 0 \;\Rightarrow\; F_\text{net} < 0$ A force opposing the motion (braking, drag, or re-entry gravity) reduces the momentum. Result: $\ev{\text{phase 1: constant thrust} \;|\; \text{phase 2: isolated, } p \text{ constant} \;|\; \text{phase 3: } dp/dt < 0}$

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Topics: Quantità di moto e urti Concepts: Quantità di moto · Impulso Skills: Lettura dei grafici Objects: Razzo