Question bank — Thrust coefficient C_F = F - (P_c A - ) — derivation

False. $T_c$ appears in both $\dot{m}$ (as $1/\sqrt{R{T}_c}$) and $u_e$ (as $\sqrt{T_c}$), and in the product $\dot{m}{u}_e/(P_c{A}^{\ast })$ the $\sqrt{R{T}_c}$ cancels; hotter gas raises thrust only through $\dot{m}$ and $c^*$, leaving $C_F$ unchanged.

True. Pressure × area is a force, so $F/(P_c{A}^{\ast })$ is force/force — a pure number, an amplification factor.

True. Sea level subtracts $P_a{A}_e/(P_c{A}^{\ast })$ from the pressure term; setting $P_a=0$ removes that penalty (or adds a bonus), so vacuum $C_F$ is strictly larger.

True. The factor $(P_e-P_a)$ becomes zero, killing the whole pressure term, leaving only $C_{F,\text{mom}}$.

False in general. The ceiling depends on $\gamma$: with small $\gamma$ (e.g. $1.2$) $C_{F,\max }\approx 2.25$; large expansion in vacuum can push practical values above $2.0$.

False. In vacuum bigger area always helps, but at sea level a very large $A_e$ can make the nozzle over-expanded ($P_e<P_a$), so the negative pressure term drops $C_F$.

False. After the $\sqrt{R{T}_c}$ cancellation only $\gamma$ and the pressure ratio $P_e/P_c$ survive; $R$ has vanished entirely.

True. With $P_e=0$ and $P_a=0$ the term $(P_e-P_a)A_e/(P_c{A}^{\ast })=0$, and $C_F$ reaches its purely momentum-set maximum $C_{F,\max }$.

True. $C_F$ knows only ratios and geometry; the propellant difference shows up in $c^{\ast }$ and $\dot{m}$, not in the nozzle report card.

The cap is on $C_F$, not on $F$. Thrust scales with $P_c{A}^{\ast }$; a bigger throat or higher chamber pressure raises $F$ freely even at fixed $C_F$.

The term is $(P_e-P_a)$: when over-expanded ($P_e<P_a$) ambient pushes back, making it negative and cutting thrust.

$P_c$ must be the stagnation pressure (gas essentially at rest). Mixing static and stagnation values breaks the isentropic relations used to get $u_e$.

$u_e$ does carry $\sqrt{T_c}$, but so does $\dot{m}$ through $1/\sqrt{R{T}_c}$; in the product $\dot{m}{u}_e/(P_c{A}^{\ast })$ the $\sqrt{R{T}_c}$ cancels, so $C_F$ has no $T_c$.

Not impossible — a well-expanded, slightly under-expanded engine ($P_e>P_a$) gains a positive pressure term and can read high; the number flags a well-matched nozzle, not an error.

They are complementary: [[Characteristic Velocity c-star|$c^{\ast }$]] grades the combustion/propellant, $C_F$ grades the nozzle. Their product gives $F=C_F{c}^{\ast }\dot{m}$.

The throat fixes $\dot{m}$ via choked flow, but the momentum term still depends on $P_e/P_c$ (how much the flow expands downstream), and the pressure term needs $P_e,P_a,A_e$.

Because once flow is choked, $\dot{m}$ is set entirely at the throat; $A^{\ast }$ is the natural "size" of the engine's mass-flow capacity, making $P_c{A}^{\ast }$ a clean reference force.

It isolates how well the nozzle shape and expansion convert chamber pressure into directed thrust, independent of how hot or heavy the propellant is.

Energy conservation ties exit speed to the pressure ratio raised to $(\gamma -1)/\gamma$, and after dividing by $P_c{A}^{\ast }$ every dimensional quantity ($R$, $T_c$) cancels, leaving only the ratio and $\gamma$.

In vacuum the pressure term $\frac{P_e}{P_c}\frac{A_e}{A^{\ast }}$ always adds, so more area gives more $C_F$; at sea level too much area over-expands and ambient pushes back.

Larger $\gamma$ means the gas releases less of its internal energy per expansion (steeper enthalpy drop needed), so less enthalpy converts to directed kinetic energy, capping $C_{F,\max }$ lower. See Nozzle Expansion Ratio Ae-over-Astar for how area sets $P_e$.

Being dimensionless, it strips out the raw scale ($P_c$, $A^{\ast }$) and leaves a pure efficiency grade — a tiny thruster and a giant booster can be compared directly.

Thrust itself is $\dot{m}{u}_e+(P_e-P_a)A_e$; dividing each piece by $P_c{A}^{\ast }$ preserves that split, giving one term from mass-motion and one from unbalanced exit pressure.

The pressure term is zero ($P_e=P_a$), and for a given $P_c/P_a$ this yields the maximum thrust — the nozzle is perfectly matched to ambient.

Yes. $(P_e-P_a)<0$ makes the pressure term negative, so $C_F=C_{F,\text{mom}}+(\text{negative})<C_{F,\text{mom}}$.

It approaches $C_{F,\max }=\sqrt{\frac{2{\gamma }^2}{\gamma -1}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}}$, a hard ceiling fixed only by $\gamma$.

The bracket $1-(P_e/P_c{)}^{(\gamma -1)/\gamma }\to 0$, so $C_{F,\text{mom}}\to 0$ — with no expansion the gas gains no directed speed and the nozzle produces no momentum thrust.

Only in extreme over-expansion where the large negative pressure term overwhelms a small momentum term (flow separation — see the definition below — usually intervenes first), so in practice $C_F$ stays positive.

No. The throat sets $\dot{m}$; the acceleration to high speed (and hence most thrust) happens in the diverging section downstream where the gas expands. See Specific Impulse Isp for how $C_F$ ties back to overall efficiency.