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I have been trying to understand the physics behind a rocket's acceleration at launch for a while now and by this point am more confused than ever. As net force is $F = ma$ --> $a =\frac Fm$. In a simplified model, the major force acting on a rocket would be the thrust. The thrust a rocket produces from the expulsion of gases is constant, so the force acting is constant. However, as mass is decreasing at a linear rate, so that force remains a constant $a$ must increase. This however, suggests a linear increase in acceleration, whereas the diagrams like the one below suggest an increasing rate of acceleration. Is my analysis above incorrect, or is there an element (like drag, gravity) that I've neglected to consider that are causing this change?

Also, I realise that the below diagram is of g-forces, but as $g$-force = $\frac{a+g}{g}$, can we say that this is going to be proportional to the force acting (i.e. if the $g$-force is parabolic then the net force is parabolic?)

Sorry if this all seems a little confused. If it does, the questions I'm asking are fundamentally:

a) Is the acceleration of a rocket linear or parabolic?

b) Can we make the above link between g-forces and actual forces acting?

enter image description here

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    $\begingroup$ Mass is in the denominator, so a linear change in mass results in a hyperbolic change in acceleration. $\endgroup$ – safesphere Jul 4 '18 at 3:34
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    $\begingroup$ "the major force acting on a rocket would be $F=ma$" The biggest force would be 'thrust', and the next largest would be weight. And the weight is certainly large enough that you can't neglect it, which means that you should be thinking in terms of $F_\text{net} = ma$. In a sense this is a minor complaint, but in my experience learning to talk about the physics correctly is strongly correlated with learning to apply the physical principles successfully. $\endgroup$ – dmckee Jul 4 '18 at 3:51
  • $\begingroup$ Oh yeah of course. No idea why I said that actually, thanks for pointing it out. And I'd say that that's a pretty fair complaint about my wording, because it's honestly just wrong. I'll edit it now. $\endgroup$ – Etched Jul 4 '18 at 3:59
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    $\begingroup$ Have you considered that $F=ma$ is wrong here since the system is losing mass? $\endgroup$ – Kyle Kanos Jul 4 '18 at 10:46
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    $\begingroup$ No, it's wrong independent of the mathematics used. $F=ma$ is valid for constant mass systems; once you lose mass, then it's not valid. $\endgroup$ – Kyle Kanos Jul 4 '18 at 11:26
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The graph in the question is from a NASA historical article on the effects of launch on humans. The plot shows sensed acceleration. Gravity cannot be sensed (locally). Accelerometers and human bodies qualify as local experiments, so gravitation is not included in that plot.

The other real forces acting on the astronauts are thrust from the rocket and atmospheric drag. Drag is rather small force for large rockets such as the Saturn V, so that can be ignored. Sans throttling or cutting off flow to a thruster, thrust and mass flow rate are more or less constant for a given stage. Given these simplifying assumptions, sensed acceleration is approximately $$a_\text{sensed} = \frac {F_{thrust}}{m(t)} = \frac {F_{thrust}}{m_0 - \dot m\,t} \tag{1}$$ where $t$ is time since launch.

Is the acceleration of a rocket linear or parabolic?

Neither. Equation (1) above is a hyperbola rather than a parabola.

Also, I realise that the below diagram is of g-forces, but as g-force = $\frac{a+g}g$, ... Can we make the above link between g-forces and actual forces acting?

No, for two reasons. The plot is of sensed acceleration. Gravitational acceleration is not included. Second, the acceleration of the rocket with respect to the Earth is the vectorial sum of the sensed acceleration and the gravitational acceleration. At launch, the Saturn V was oriented vertically. At first stage cutoff, the vehicle was flying much closer to horizontal than vertical.

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