M020 · Thrust, gravity and empty hardware

Variable-Mass Rocket & Staging

Does another rocket stage always produce more velocity? No. Discarding spent structure leaves less mass for the next burn, but another engine and separation system also add mass at the start. At fixed payload and propellant, enough extra hardware can make more stages deliver less upward velocity. Separation itself supplies no velocity impulse in this ideal model.

Change mass flow, exhaust speed, throttle and staging. Compare thrust, mass ratio, gravity loss and the extra hardware carried by each design.

Thrust $3.52\times10^{4}\,\mathrm{N}$

Active vehicle mass $790\,\mathrm{kg}$

Propulsive mass ratio $1.48$

Propellant velocity contribution $860\,\mathrm{m\,s^{-1}}$

Actual upward velocity change $628\,\mathrm{m\,s^{-1}}$

Gravity velocity deficit $232\,\mathrm{m\,s^{-1}}$

Model assumptions

One-dimensional point-mass flight starts momentarily at rest, already free of supports, in a uniform downward gravity field or with gravity off. There is no ground surface, atmospheric drag, orbital curvature or planet model. Exhaust leaves axially at the controlled speed relative to the craft; pressure thrust and engine transients are omitted. Full-throttle flow multiplied by throttle gives actual outflow. Throttle is off or at least one percent. Each stage carries a structural mass proportional to its propellant plus one fixed hardware overhead; comparisons hold payload and total propellant fixed. At burnout, lower dry stages leave with the current velocity and no separation impulse; the final dry stage remains attached. The propulsive mass ratio multiplies the burn-segment ratios and excludes mass discarded without thrust. Velocity loss from gravity includes the short coast after burnout. Flight displacement is compressed relative to enlarged hardware; exhaust particles indicate flow rather than individually tracked gas. Tank dimensions and fuel-volume display are schematic. Chemical energy, nozzle pressure, separation mechanisms and collisions with exhaust or discarded stages are not simulated. Sliders recompute a trial at the inspected time; dragging a tank prepares a fresh load.

Physics tutorial

Carry less through the next burn

BackgroundA rocket changes its own mass as it expels propellant. The same thrust accelerates the remaining vehicle more strongly near the end of a burn. Discarding a spent lower stage helps the next burn, while extra engines and separation hardware add mass at the start.

Why it mattersNASA Glenn’s ideal rocket equation and thrust equation, together with Peraire and Widnall’s MIT Lecture L14, establish the mass-flow balance used here. The staging structure rule is an explicit teaching choice.

Start with the essentials

Focus question: With the same propellant and payload, when is another stage worth its hardware?
One-sentence intuition: Each burn accelerates only the mass still aboard. Count a stage’s dry hardware both when carrying it and when discarding it.

Core mathematical model

Mass leaves; momentum leaves with it

\dot M=-q,\qquad T=qu_e,\qquad M\dot v=T-Mg

Upward velocity is positive. Outflow is the full-throttle mass flow multiplied by the throttle fraction. Relative exhaust speed is fixed; pressure thrust is omitted.

One powered interval

\Delta v_{\rm prop}=u_e\ln\frac{M_i}{M_f},\qquad v-v_i=\Delta v_{\rm prop}-g\Delta t

Each powered interval starts with the velocity left by the previous interval. The full trial starts momentarily at rest; gravity also continues through the final coast.

Multiply burn ratios, not discarded mass

R_{\rm prop}=\prod_i\frac{M_{i,+}}{M_{i,-}},\qquad\Delta v_{\rm prop}=u_e\ln R_{\rm prop}

The masses bracket each burn before dry-stage release. A discarded stage leaves with the current velocity, so that event adds no velocity impulse.

A fair hardware comparison

D_i=\epsilon P_i+D,\qquad M_0=M_p+(1+\epsilon)P+nD

The comparison fixes payload, total propellant, exhaust speed, flow and throttle. Each lower stage takes the chosen fraction of remaining propellant; the last takes the rest.

The moving control volume

\frac{d(Mv)}{dt}=-Mg-q(v-u_e)

The exhaust velocity here is measured in the inertial frame. Including expelled gas and discarded dry stages restores the momentum balance of the original total mass.

Common difficulties

Splitting a burn is not automatically staging

Typical misconceptionMore firing intervals must always produce more velocity.

Better mental modelWith massless stage hardware and the same exhaust speed, the burn ratios telescope to the same result. A benefit appears when dry mass can leave before later burns.

Vacuum and gravity are separate choices

Typical misconceptionRemoving air removes gravity loss.

Better mental modelBoth presets omit air. Only turning gravity off removes the downward external force; a slower burn under gravity accumulates a larger velocity deficit.

Run the experiment

01
Watch an empty stage depart
Use Let the empty stage go, then pause near the first separation.
What to observe: The active mass falls abruptly while velocity remains continuous. The next burn accelerates the lighter craft.
02
Charge each stage for its hardware
Choose Extra stages cost mass and compare the three table rows.
What to observe: Added hardware can make three stages deliver less upward velocity than one at the same propellant and payload.
03
Give gravity more time
Compare Give gravity more time with Turn gravity off.
What to observe: The propellant contribution depends on burn mass ratios; the gravity deficit depends on elapsed time.