Effective exhaust velocity c = v_e + (P_e − P_a)A_e - ṁ

WHAT is $c$ ?

HOW: Derive thrust from first principles

Step 1 — Momentum thrust (Newton's 2nd/3rd law). Take the rocket as our system. In time $dt$ it ejects mass $dm = \dot m\, dt$ backward at speed $v_e$ (relative to rocket).

Why this step? Force = rate of change of momentum. The gas gains rearward momentum, so by reaction the rocket gains equal forward momentum.

$F_{\text{mom}} = \frac{d p_{\text{gas}}}{dt} = \frac{v_e\,dm}{dt} = \dot m\, v_e$

Step 2 — Pressure thrust (a force balance on the gas at exit). Consider the exit plane, area $A_e$ . Inside the exit the gas pushes outward with pressure $P_e$ ; outside, ambient air pushes inward with $P_a$ .

Why this step? Pressure is force per unit area. Over the exit ring the net outward push is $(P_e - P_a)$ acting on $A_e$ . This force is not cancelled by anything (there's no wall there), so it becomes thrust.

$F_{\text{pres}} = (P_e - P_a)\,A_e$

Step 3 — Add them.

$F = \underbrace{\dot m\, v_e}_{\text{momentum}} + \underbrace{(P_e - P_a)A_e}_{\text{pressure}}$

Step 4 — Absorb everything into one velocity $c$ . We define $c$ so that $F = \dot m\, c$ . Divide the whole thing by $\dot m$ :

$c = \frac{F}{\dot m} = v_e + \frac{(P_e - P_a)A_e}{\dot m}$

Why bother? Because now every rocket equation ( $F=\dot m c$ , $I_{sp}=c/g_0$ , Tsiolkovsky $\Delta v = c\ln(m_0/m_f)$ ) uses ONE number.

Figure — Effective exhaust velocity c = v_e + (P_e − P_a)A_e - ṁ

Special cases (Forecast-then-Verify)

Worked example Worked Example 1 — Sea level vs vacuum

A nozzle has $\dot m = 200$ kg/s, $v_e = 2800$ m/s, $P_e = 70$ kPa, $A_e = 1.2$ m². (a) At sea level $P_a = 101.3$ kPa: $c = 2800 + \frac{(70000-101300)(1.2)}{200} = 2800 + \frac{-37560}{200} = 2800 - 187.8 = 2612\ \text{m/s}$ Why negative? $P_e<P_a$ → overexpanded → thrust penalty. (b) In vacuum $P_a = 0$ : $c = 2800 + \frac{(70000)(1.2)}{200} = 2800 + 420 = 3220\ \text{m/s}$ Why this step? Setting $P_a=0$ removes the back-push; the full exit pressure now helps. Same engine gives more $c$ in space — this is why $I_{sp}$ is quoted higher in vacuum.

Common mistakes

Active Recall

Recall Quick self-test (hide and answer)

Why does a rocket produce pressure thrust?
When is $c = v_e$ exactly?
Does $c$ go up or down as the rocket climbs? Why?
What is the sign of the correction if $P_e < P_a$ , and what is that condition called?

Recall Feynman: explain to a 12-year-old

Imagine blowing up a balloon and letting it go. The air rushing out backward shoves the balloon forward — that's the "speedy gas" push. Now imagine the balloon's mouth is a bit narrow so the air inside is still squeezed harder than the room's air right at the opening — that extra squeeze also shoves the balloon a little more. A rocket has both pushes. Instead of tracking two shoves, scientists pretend the gas came out at ONE make-believe speed, $c$ , that gives the exact same total push. High up in space there's no room-air pushing back, so the make-believe speed gets bigger and the rocket pushes harder!

Flashcards

Define effective exhaust velocity

c

The single equivalent speed satisfying

F=\dot m c

, including momentum and pressure thrust:

c = v_e + (P_e-P_a)A_e/\dot m

Write the full thrust equation before defining

c

F = \dot m v_e + (P_e - P_a)A_e

When does

c = v_e

When the nozzle is optimally expanded, i.e.

P_e = P_a

(pressure term vanishes).

Sign of the pressure term when underexpanded (

P_e>P_a

Positive →

c>v_e

(extra thrust).

Sign when overexpanded (

P_e<P_a

Negative →

c<v_e

(thrust penalty, risk of flow separation).

Why does

c

increase with altitude for the same engine?

Because

P_a

decreases, making

-P_aA_e/\dot m

less negative; the exit pressure's push is opposed less by ambient air.

Where does pressure thrust physically come from?

The net outward force

(P_e-P_a)

acting over the exit area

A_e

, which has no wall to cancel it.

Relation between

c

and specific impulse?

I_{sp} = c/g_0

, so

c = I_{sp}\,g_0

Given

c

, how do you recover raw exit velocity

v_e

v_e = c - (P_e-P_a)A_e/\dot m

Connections

Thrust Equation — this note's parent; $F=\dot m v_e + (P_e-P_a)A_e$ .
Specific Impulse — $I_{sp}=c/g_0$ ; effective velocity in disguise.
Tsiolkovsky Rocket Equation — uses $c$ : $\Delta v = c\ln(m_0/m_f)$ .
Nozzle Expansion (Under/Over/Optimal) — sets the sign of the pressure term.
Conservation of Momentum — origin of momentum thrust.
Atmospheric Pressure vs Altitude — why $c$ varies during ascent.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, rocket ko do tarah se thrust milta hai. Pehla hai momentum thrust — gas peeche ki taraf tez speed $v_e$ pe nikalti hai, aur Newton ke third law se rocket ko aage dhakka milta hai, jiska maan $\dot m v_e$ hota hai. Doosra hai pressure thrust — nozzle ke exit pe gas ka pressure $P_e$ aur bahar ki hawa ka pressure $P_a$ same nahi hote. Is difference $(P_e - P_a)$ ko exit area $A_e$ se multiply karo, aur wo bhi ek real force banta hai jo thrust me add hota hai.

In dono ko alag-alag track karna jhanjhat hai, isliye scientists ne ek smart trick nikali: ek "as if" velocity define kar di jise effective exhaust velocity $c$ kehte hain, taaki thrust simply $F = \dot m c$ ban jaye. Formula hai $c = v_e + (P_e - P_a)A_e/\dot m$ . Jab nozzle perfect ho ( $P_e = P_a$ ), tab pressure term zero, aur $c = v_e$ .

Sabse important baat: jaise-jaise rocket upar jaata hai, bahar ka $P_a$ kam hota jaata hai (space me toh zero). Isliye $-P_a A_e/\dot m$ wala penalty kam ho jaata hai aur $c$ badh jaata hai. Yahi wajah hai ki same engine ka vacuum $I_{sp}$ sea-level se zyada hota hai. Exam me galti mat karna — $c$ aur $v_e$ ko blindly barabar mat maano, pehle check karo ki $P_e = P_a$ hai ya nahi.

Effective exhaust velocity c = v_e + (P_e − P_a)A_e - ṁ

WHAT is $c$ ?

HOW: Derive thrust from first principles

Special cases (Forecast-then-Verify)

Common mistakes

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Rocket Propulsion

Connections