Acceleration of a Rocket at Launch

Question

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?

Answer 1

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.