Now, remember Newton’s third law: for every action there is an equal and opposite reaction. It means the act of ejecting molecules pushes back on the rocket: every time a water molecule flies from the nozzle, the rocket loses a little mass and in return gains a little velocity. If the rocket burns fuel at a high rate, the reaction force on the rocket will be high, and so will the acceleration. The force is approximately proportional to the rate at which propellant is consumed.

Each type of chemical rocket has its own chemistry (liquid hydrogen and liquid oxygen; kerosene and liquid oxygen; solid fuel; etc.), and that chemistry combined with the design of the engine results in a characteristic velocity for the ejected molecules, called the exhaust velocity. (Technically it’s the “effective exhaust velocity,” but I’ll shorten it to “exhaust velocity” in an attempt at brevity. It’s also very closely related to the “specific impulse” which I won’t cover here but you can easily look up if you’re interested.) In the case of burning hydrogen, if all the energy released in the chemical reaction were perfectly and uniformly transferred into kinetic energy of the resulting water molecules, they would fly from the engine at about 5200 m/s. A real engine can’t achieve that velocity because some energy is inevitably lost to heat and light, and some of the molecules leave the engine traveling somewhat to the side, not straight backward, so the conversion of energy into velocity isn’t 100% efficient. The exhaust velocity for the RS-25 and RL-10 engines is about 4400 m/s in vacuum, 85% of the theoretical value.

Our goal is to figure out how the mass of propellant, the rocket, and the payload are related to the final velocity after the propellant is used up. We need a way to relate the change in mass to the change in velocity, and to do that we’ll use one of the most fundamental of all physical laws: the conservation of momentum. Momentum is simply the product of mass times velocity, and its direction matters, which means two momenta can cancel each other if they are pointed in opposite directions. In the figure below, in the left panel there is a rocket with total mass m at rest, with zero velocity and hence zero momentum. In the right panel, the rocket has ejected a small mass dm with velocity ve, the exhaust velocity, to the left, and as a result the rocket moves with a small velocity dv in the opposite direction, to the right.