Propulsion: Throwing Mass Backwards

Blow up a balloon, pinch the neck, let go. It shoots around the room, and nothing is pushing against anything: there is no ground in the air for it to shove off. A fire hose is the same idea with muscle. Firefighters brace themselves because the water leaving the nozzle shoves the hose backwards.

That is the whole of propulsion. A jet engine, a rocket and an ion thruster are the same machine at three scales: throw mass backwards, and the machine goes forwards. The rest of the subject is about how to do that cheaply, with the least fuel for the push you need. The answers are surprising: an airliner wants huge, gentle exhaust, a rocket wants a hot, fast one, and a satellite wants a nearly invisible one.

Thrust is momentum flux: push mass backwards

Newton's second law says force equals the rate at which momentum changes. An engine takes in some mass each second, speeds it up, and shoves it out the back. The momentum it adds per second is the force on the engine. For a rocket standing still, it is as simple as it gets:

Notice you can get the same thrust from many different recipes. A lot of mass thrown slowly, or a little mass thrown very fast. They are not equal in cost, though. Making the exhaust move costs kinetic energy, ½ ṁ v_e², and that grows with the square of the speed. Slide the exhaust speed below for a fixed 100 kN push.

At 1,000 m/s you throw 100 kg a second and pay 50 MW. At 10,000 m/s you throw only 10 kg a second, but the exhaust carries away 500 MW. Fast exhaust saves propellant and costs power. Which one matters more depends on whether you are carrying the propellant (a rocket) or scooping it from the air for free (a jet).

Propulsive efficiency: why airliners use big, slow jets

A moving vehicle has a second problem: the exhaust is not wasted only if it ends up still. If a jet flies at 250 m/s and throws gas back at 600 m/s, the gas leaves the scene still moving at 350 m/s relative to the ground. That leftover kinetic energy is lost. The fraction of the engine's energy that does useful pushing is the propulsive efficiency:

So at the same thrust, the best engine for a slow vehicle moves a large amount of air a small amount faster. That is the logic of the propeller: a huge disc that grabs a lot of air and speeds it up gently. It is also the logic of the turbofan. A jet engine's core makes hot gas. Instead of sending it all out the back, most of its power drives a big fan at the front, which pushes a stream of air around the core. The bypass ratio is the mass of air going around the core divided by the mass going through it.

At 250 m/s (about 900 km/h, an airliner's cruise), a bare core gives 17.5 kN at 59% efficiency. Bypass ratio 8, with the same core, gives about 26.6 kN at 89%: more push for the same fuel. Now raise the flight speed towards 400 m/s: the advantage shrinks, and the fan, which is fat and drag-heavy, stops paying for itself. That is why airliners cruise at high subsonic speed with high-bypass fans and fighters use thin, low-bypass engines. Published ratios run roughly 0.3 to 0.7 for combat engines and about 5 to 12 for airliners: the LEAP-1A is around 11:1, the GE9X around 10:1 and the PW1000G about 12:1.

Inside a jet: the Brayton cycle and the engine family

The core of every gas-turbine engine is the Brayton cycle you met in Thermodynamics (34/14): a compressor squeezes the air, a combustor burns fuel in it continuously, and a turbine takes some of the hot gas's energy to turn the compressor. What is left either shoots out as a jet or turns a fan or a propeller. The ideal efficiency 1 − (pressure ratio)^−0.286 rises with pressure ratio, which is why engines push the compressor harder and the combustor hotter. The turbine runs in gas hotter than its blade metal would survive uncooled, which is where Heat Transfer (34/16) comes in.

Engine	What gives the push	Sweet spot
Turboprop	Core drives a propeller through a gearbox	Slower, lower-flying aircraft
High-bypass turbofan	Mostly the big fan, a bit of core jet	Airliners, just under the speed of sound
Low-bypass turbofan, turbojet	Mostly the fast core jet (often with afterburner)	Fighters, supersonic flight
Ramjet	No compressor: speed squeezes the air	Roughly Mach 3 to 6, needs to be moving already
Scramjet	Burns fuel in air that stays supersonic	From about Mach 5 or 6 upward

Take away the compressor and the turbine, and let speed do the squeezing. That is a ramjet. Slow the incoming air to a stop and you recover its kinetic energy as pressure and heat.

At Mach 3 the ideal pressure ratio is about 37 to 1, as good as a big compressor, with no moving parts. At Mach 5 slowing the air fully would heat it to about 1,300 K (over 1,000 °C), and faster still would heat it until its molecules start to break apart. That is why a scramjet keeps the air supersonic while it burns fuel. Scramjets are hard: NASA's X-43A reached about Mach 9.6 on 16 November 2004, the record for an air-breathing aircraft, in a flight lasting seconds. Both ramjets and scramjets produce no thrust standing still, so they need a rocket or another engine to get them up to speed.

The rocket equation: why rockets come in stages

A rocket carries its own propellant, and that is the catch. To speed up, it has to accelerate its fuel too, and the fuel it has not yet burned is dead weight until it does. Konstantin Tsiolkovsky wrote the answer down in 1903:

The logarithm is the bad news. To add a fixed amount of speed you must multiply the starting mass by the same factor each time. With an exhaust speed of 3.4 km/s (Isp 350 s), every extra 3.4 km/s takes you from a mass ratio of e (2.7) to e² (7.4) to e³ (20). Reaching low orbit takes about 9.4 km/s once you include the energy lost to gravity and drag; published figures run about 9 to 10 km/s depending on the trajectory. A single tank-and-engine would need almost all of its mass to be propellant.

The fix is to drop the empty tanks. When the first stage is empty, you throw its engines and tank away and continue with a lighter rocket, so the remaining fuel has less mass to shove. All orbital launchers in use today work this way. Try it.

With the defaults, one stage reaches 7.3 km/s: short of orbit. Two stages reach 10.7 km/s. The same payload fraction and the same engines; the only change is throwing the empty tanks away. Now drag the structure slider up to 20%: a single stage cannot get anywhere near. Rocket designers fight for every percent of structure for exactly this reason.

Specific impulse: miles per gallon for rockets

A car has miles per gallon. A rocket has specific impulse: the thrust you get per unit weight of propellant burnt each second. Divide thrust by the propellant's weight flow and the answer comes out in seconds.

It is the exhaust speed in disguise: an Isp of 450 s means an exhaust speed of about 4.4 km/s. What sets it is the propellant and the engine. Hot, light exhaust is fastest, so hydrogen burnt with oxygen leads the chemical rockets.

• Solid rockets (propellant cast like a rubbery block): simple, storable, enormous thrust, roughly 250 to 270 s. Once lit they essentially cannot be throttled or shut down at will, so they work as boosters and missiles. The Space Shuttle's solid boosters were about 268 s in vacuum.
• Liquid kerosene and oxygen: roughly 300 to 350 s in vacuum. SpaceX's Merlin 1D is about 311 s in its sea-level form and about 348 s in the vacuum version, per published specs, which vary by source.
• Liquid hydrogen and oxygen: about 450 s. The Space Shuttle's RS-25 engine is rated 452 s in vacuum and 366 s at sea level. Hydrogen is very light, so the tanks are huge.
• Air-breathing jets do not carry their oxidiser, so counted the same way their Isp is several thousand seconds, which is why jets beat rockets inside the atmosphere.

Sea level versus vacuum differ because the atmosphere pushes back on the nozzle; the same engine does better in space. Slide the Isp below and watch what a mission costs.

For a 4 km/s speed change on a craft that ends at 1,000 kg: Isp 250 s needs about 4,110 kg of propellant, Isp 450 s needs 1,480 kg, and Isp 3,100 s needs 141 kg. That is why people care about every second of Isp.

Electric propulsion: tiny thrust, huge Isp

If exhaust speed is what matters, use electricity instead of chemistry. An ion thruster ionises a gas such as xenon and accelerates the ions through an electric field to tens of kilometres per second. NASA's NSTAR thruster, first flown on Deep Space 1 (launched 24 October 1998), reaches an Isp above 3,100 s with a maximum thrust of 92 mN at 2.3 kW of power. That is the weight of a couple of sheets of paper. The exhaust is 10 times faster than a chemical rocket's, and the propellant needed is a tenth as much. The price is time. With only a few kilowatts of solar power available, the push is tiny, so the thruster must run for months.

With 100 kg of propellant the ion thruster gives a speed change of about 3.2 km/s, ten times the chemical thruster's 0.33 km/s, but it takes about 380 days of continuous thrusting, against under 12 minutes. The acceleration is about a hundred-thousandth of a metre per second squared. Electric propulsion is a long-haul tool: station-keeping, raising orbits and deep-space probes. It cannot lift anything off the ground. Check the arithmetic: thrust power ½ F v_e is about 1.4 kW out of 2.3 kW in, a thruster efficiency of about 61%, in line with NASA's stated figure of above 61%.

What a graduate just sees

• Every vehicle moves by throwing something backwards. Thrust is mass flow times velocity change.
• The same push can come from a lot of slow mass or a little fast mass, and the kinetic-energy bill climbs with the square of the speed.
• A vehicle wastes energy when its exhaust is much faster than itself. That is why airliners use huge fans and propellers use huge discs.
• A jet engine is a Brayton cycle in a tube. Pressure ratio and turbine temperature are the two numbers that matter.
• A rocket is mostly propellant, because the rocket equation is a logarithm: each extra bit of speed multiplies the mass.
• Staging works by throwing away dead weight at the right time.
• Isp is the exhaust speed in seconds. A higher Isp is a smaller tank for the same journey.
• Thrust and Isp are different goals. Chemical engines trade Isp for thrust; ion engines do the opposite.

Where it shows up next

• Systems Engineering & the Capstone (34/22): engine mass, propellant and structure fight each other in a vehicle design, and the rocket equation is the referee.
• Heat Transfer (34/16), Machine Design (34/19): turbine blades, nozzles and bearings are heat transfer and machine design under extreme conditions.
• Control Systems (34/20): engines are throttled, gimballed and steered by feedback.
• In the world: the LEAP and GE9X engines on airliners, reusable first stages, GPS and communication satellites that keep station with electric thrusters, and any argument about what a fuel or an engine "should" cost per kilometre.

If you want to go deeper

• 📘 Textbook. Jack Mattingly and Keith Boyer, Elements of Propulsion: Gas Turbines and Rockets (AIAA, 2nd edition, 2016). A standard undergraduate text for both jets and rockets.
• 📘 Textbook. George Sutton and Oscar Biblarz, Rocket Propulsion Elements (Wiley, 9th edition). The classic for liquid, solid and electric rockets.
• 📘 Gentler. John D. Clark, Ignition! An Informal History of Liquid Rocket Propellants (Rutgers University Press, 1972; reissued 2018): a chemist's funny, vivid account of how rocket fuels were found. Or the library's own Rockets and Spaceflight (29/06) for the history.
• 🌐 Free course. MIT OpenCourseWare 16.01 Unified Engineering: Thermodynamics and Propulsion (Fall 2005): thrust, efficiency, rocket performance and ideal cycle analysis of jet engines. For rockets in depth, 16.512 Rocket Propulsion (graduate).