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Exhaust speed

Rockets and landing · level 5, lesson 5 of 6

A question to start

One engine burns 300 kg of propellant a second and gives 850 kN of thrust. Another burns only 25 kg a second and gives 110 kN. Which is the better engine for getting to Mars?

The idea

Each second, an engine throws out a mass \(\dot m\) (said "m-dot", the mass flow rate, in kg/s) at the exhaust speed \(v_e\) relative to the rocket. The momentum it gives the exhaust each second is \(\dot m v_e\), and by Newton's second and third laws, that is the thrust:

\[T = \dot m\, v_e.\]

(There is a small extra term from the pressure of the exhaust at the nozzle's exit, which this lesson leaves out. It is why engines give more thrust in a vacuum than at sea level.)

So thrust can come from a lot of mass thrown slowly or a little mass thrown fast. For lifting off the ground, you need a big thrust, so a big flow. But for going far on a fixed load of propellant, what matters is \(v_e\): each kilogram of propellant gives more push for longer when it is thrown faster.

Engine Thrust Mass flow Exhaust speed \(v_e = T/\dot m\)
Merlin 1D, sea level (kerosene and oxygen) about 845 kN about 300 kg/s about 2.8 km/s
F-1, Saturn V first stage, sea level about 6,770 kN about 2,580 kg/s about 2.6 km/s
RL10 (hydrogen and oxygen, in vacuum) about 110 kN about 24 kg/s about 4.5 km/s
Ion thruster (Dawn spacecraft) 0.09 N about 3 mg/s about 30 km/s

The RL10 burns a twelfth of the Merlin's flow, but each kilogram does 60% more work. The ion thruster's push is about the weight of a sheet of paper, but its exhaust is ten times faster again: tiny thrust, enormous efficiency.

The best engine for Mars? For leaving Earth's surface, a high-thrust engine. For the trip, the higher the \(v_e\), the less propellant is needed. Level 6 makes this exact with the rocket equation.

Worked examples

1. The engines in the question. First: \(v_e = 850{,}000/300 = 2{,}830\) m/s. Second: \(110{,}000/25 = 4{,}400\) m/s. The second is better for the trip; the first for lift-off.

2. Propellant for a minute. An RL10 firing for 60 s uses \(24 \times 60 = 1{,}440\) kg.

3. A Mars lander's engines. To hover, a 1,000 kg lander needs 3,710 N. With \(v_e = 2{,}200\) m/s (a simple hydrazine engine), \(\dot m = 3710/2200 = 1.7\) kg/s.

Try it

  1. An engine gives 200 kN with a mass flow of 70 kg/s. Find \(v_e\).
  2. What mass flow gives 1,000 kN at \(v_e = 3{,}300\) m/s?
  3. How much propellant does the lander in example 3 use if it hovers for 20 s?
  4. Why do engines give more thrust in a vacuum than at sea level?
  5. Why can't an ion thruster launch a rocket from Earth?
Answers
  1. \(200{,}000/70 = 2{,}860\) m/s.
  2. \(\dot m = 1{,}000{,}000/3300 = 303\) kg/s.
  3. \(1.7 \times 20 = 34\) kg.
  4. The air outside presses back on the exhaust at the nozzle's exit; with no air, there is nothing pushing back.
  5. Its thrust is tiny, far less than its own weight: TWR is about \(10^{-5}\).

Common mistake

Judging an engine by thrust alone. Thrust says how hard it pushes now; exhaust speed says how much push you get from each kilogram of propellant. For long trips, exhaust speed matters more.

Where it leads