Rocket Science: propulsion and trajectory maths

01

Where thrust comes from: the thrust equation

In simple words

Throw a heavy ball off a skateboard and you roll backwards. Throw more balls each second, or throw them faster, and you roll back harder. A rocket engine throws gas out of its nozzle instead of balls: more gas each second, thrown faster, means more push.

Before this course: Model Rocket Projects: design and flight (course 15.1.2). You already know the four forces, motor codes, stability, simulation with OpenRocket and RocketModeler, recovery and the launch-day routine. This course adds the maths behind those flights, so you can predict them yourself and check the computer.

Key facts

The rocket thrust equation (NASA Glenn): F = ṁ x Ve + (pe - p0) x Ae

Symbol	Meaning	Unit
F	thrust	N (newtons)
ṁ ("m-dot")	mass flow rate: kilograms of exhaust leaving each second	kg/s
Ve	exhaust speed at the nozzle exit	m/s
pe	pressure of the exhaust at the nozzle exit	Pa
p0	pressure of the air around the rocket	Pa
Ae	area of the nozzle exit	m²

The first part (ṁ x Ve) is the push from throwing mass backwards. The second part is a pressure correction. NASA Glenn notes the exit pressure only equals the outside pressure at one design condition.
Why a rocket works in space: it carries its own oxidizer, so it can make thrust in a vacuum. In a vacuum p0 = 0, so the pressure part can only add thrust. The same engine pushes a little harder in space than at sea level.
Nozzle shape sets the numbers: NASA Glenn says the nozzle throat sets the mass flow rate, and the ratio of exit area to throat area sets the exit speed and exit pressure.
The equation works for both liquid and solid rockets (NASA Glenn).
Air pressure at sea level: NASA Glenn's metric atmosphere equations give about 101 kPa (101,000 Pa) at sea level.

Example (example numbers, not a real engine)

An engine throws out ṁ = 250 kg/s at Ve = 3,000 m/s. The nozzle exit is Ae = 1.0 m² with exit pressure pe = 60,000 Pa.

Momentum part: 250 x 3,000 = 750,000 N.
At sea level (p0 = 101,000 Pa): pressure part = (60,000 - 101,000) x 1.0 = -41,000 N. Thrust = 709,000 N (709 kN).
In a vacuum (p0 = 0): pressure part = +60,000 N. Thrust = 810,000 N (810 kN).

Step by step

Write the thrust equation at the top of a page and label every symbol with its unit.
Copy the example numbers into a table: ṁ, Ve, pe, p0, Ae.
Work out the momentum part (ṁ x Ve).
Work out the pressure part at sea level, then in a vacuum.
Add the parts to get the thrust in both places.
Change only Ve to 3,300 m/s and work it out again.
Check: your sea-level thrust is 709 kN, your vacuum thrust is 810 kN, and a faster exhaust gave more thrust.

Watch on YouTube

Thrust and specific impulse equationsAerospace Engineering Explained · YouTube · youtube.com

Videos by other creators on YouTube, for reference. They are not made by or affiliated with The Future Market.

LESSON 01Where thrust comes from: the thrust equation

LESSON 01 / ANIMATED LESSON / AI VOICE00:00 / 01:00

Voice

Read the video script

[Host on a skateboard holding a ball] "Throw something backwards and you move forwards. That is a rocket in one move."
[Text on screen: F = m-dot x Ve + (pe - p0) x Ae] "Thrust is mass per second times exhaust speed, plus a pressure correction."
[Screen: same engine at sea level and in space, thrust arrows growing] "In space the outside pressure is zero, so the same engine pushes a little harder."
[Screen: nozzle with throat and exit labelled] "The nozzle's throat sets how much gas flows. The exit sets how fast it leaves."
[Host] "More mass per second, or faster exhaust, means more thrust."

Try it now

Using the example engine, make a three-row table: sea level, halfway up (p0 = 50,000 Pa) and vacuum. Work out the thrust for each row. In one sentence, explain why the thrust grows as the rocket climbs.

ClaudeStuck? Copy a ready-made prompt for Claude

Copy this, open Claude, paste it, and Claude will walk you through the lesson step by step.

For a grown-up: Claude is for adults (18+). Copy this, paste it into Claude, and use the answer to explain the lesson to your child.

I'm learning "Rocket Science: propulsion and trajectory maths" on The Future Market. Lesson 1: "Where thrust comes from: the thrust equation".

What the lesson teaches:
- The rocket thrust equation (NASA Glenn): F = ṁ x Ve + (pe - p0) x Ae
- Symbol - Meaning - Unit
- F - thrust - N (newtons)
- ṁ ("m-dot") - mass flow rate: kilograms of exhaust leaving each second - kg/s
- Ve - exhaust speed at the nozzle exit - m/s
- pe - pressure of the exhaust at the nozzle exit - Pa
- p0 - pressure of the air around the rocket - Pa

The exercise I'm working on:
Using the example engine, make a three-row table: sea level, halfway up (p0 = 50,000 Pa) and vacuum. Work out the thrust for each row. In one sentence, explain why the thrust grows as the rocket climbs.

I'm stuck. Please:
1. Explain the main idea in plain, simple words with one everyday example.
2. Walk me through the exercise one small step at a time. Ask me what I've done so far before giving the next step.
3. Don't just give me the final answer. Help me do it myself, and check my work when I share it.

I'm a parent helping my child (about 8 years old) with a lesson called "Where thrust comes from: the thrust equation" from the course "Rocket Science: propulsion and trajectory maths".

Here's what the lesson says:
Throw a heavy ball off a skateboard and you roll backwards. Throw more balls each second, or throw them faster, and you roll back harder. A rocket engine throws gas out of its nozzle instead of balls: more gas each second, thrown faster, means more push.

The activity:
Using the example engine, make a three-row table: sea level, halfway up (p0 = 50,000 Pa) and vacuum. Work out the thrust for each row. In one sentence, explain why the thrust grows as the rocket climbs.

Please explain this so an 8-year-old understands: short sentences, a fun everyday example, and no hard words.
Then give me 3 simple questions I can ask my child to check they understood, and a way to do the activity together, one tiny step at a time.

Open Claude

Sources

NASA Glenn Research Center, Rocket Thrust Equation: www1.grc.nasa.gov/beginners-guide-to-aeronautics/rocket-thrust-equatio
NASA Glenn Research Center, Earth Atmosphere Equation (Metric): www1.grc.nasa.gov/beginners-guide-to-aeronautics/earth-atmosphere-equa

My notes

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