Rocket thrust comes from momentum flow

Choose forward as positive. Let the rocket's instantaneous mass be m, velocity v, and mass derivative negative during a burn. Let u be a positive exhaust speed backward relative to the rocket. During a short interval the expelled positive mass is minus the rocket's mass change. To first order its inertial velocity is v minus u. Apply momentum conservation to the rocket and that exhaust parcel, including any external impulse. Terms involving the arbitrary starting velocity cancel. What remains is a forward thrust proportional to positive exhaust speed times the positive rate of expelled mass.

This cancellation matters physically: an observer moving at constant speed should infer the same rocket acceleration and thrust. An incorrect equation that treats the open rocket as a fixed system can produce an extra term involving its absolute velocity, violating this expectation. For other variable-mass systems, such as a cart collecting rain, the entering material has its own velocity and the momentum-flux term changes accordingly. There is no universal rule that mass loss automatically gives forward acceleration. The relative velocity and direction of the transferred material determine the momentum exchange.

For constant exhaust speed and no external force, divide the equation by the instantaneous remaining mass and integrate over the burn. The logarithm arises because expelling a given amount of propellant produces more velocity change when less mass remains. Initial mass includes propellant and structure; final mass includes the structure and any remaining payload or fuel. Both must be positive, and the final mass must be smaller during an actual burn. Using propellant mass alone as the denominator would discard the inert mass that the exhaust must also accelerate.

The ideal velocity gain depends on the mass ratio rather than the duration of the burn when the stated assumptions hold. A long slow burn and a short fast burn with equal mass ratio can therefore have the same ideal gain in free space. This does not make burn timing irrelevant near a planet or in an atmosphere. Gravity acts during the elapsed time, drag depends on trajectory and speed, and engine performance can vary. Also distinguish velocity gain from final speed: direction and initial velocity matter, and a sequence of burns in different directions cannot generally be reduced to one scalar speed addition.

For a vertical upward burn with approximately constant g, no drag, and a thrust axis kept upward, integrating the gravity term subtracts g times burn duration from the ideal gain. The instantaneous condition for upward acceleration is thrust greater than weight, not merely nonzero exhaust. A rocket can be burning while decelerating upward or accelerating downward. Atmospheric pressure differences at the nozzle can also contribute thrust; using an effective exhaust speed is a compact engineering convention only when its meaning and operating conditions are stated. Do not add a pressure correction again if it is already included in that effective value.

Audit a rocket calculation by checking three signs and one boundary. Mass must decrease during expulsion, backward exhaust must produce forward thrust, and gravity must have the sign dictated by the axis. Then verify whether the model treats the boundary as the open remaining rocket or the closed rocket-plus-exhaust collection. Kinetic energy is not conserved for the moving masses alone because stored internal energy powers the exhaust. Momentum accounting remains valid without assuming that the chemical energy conversion or nozzle flow has been modeled in detail. These distinctions keep a compact mechanics calculation separate from a full propulsion simulation.