EUFISICA · José Gonçalves

Space explorationPhysics explained

The Physics of Rockets

From Newton's laws and the rocket equation to orbital mechanics and the future of space exploration.

How can a vehicle weighing hundreds of tonnes accelerate through Earth's atmosphere, reach orbit, and travel through deep space? The answer combines conservation of momentum, thermodynamics, fluid dynamics, and orbital mechanics. In this illustrated guide, we derive the fundamental equations, examine real-world scales, and use mathematical models to investigate rocket performance.

Circular speed · 400 km altitude

7.67 km/s

Approximately 27,600 km/h

Escape speed · Earth's surface

11.2 km/s

Ideal, unpowered escape condition

01 / FUNDAMENTALSWhy do rockets work in a vacuum?

Newton's third law describes a fundamental aspect of rocket propulsion: when a rocket accelerates exhaust gases backwards, the gases transfer momentum to the rocket in the opposite direction. The rocket does not push against air. Because it carries both fuel and oxidizer (for chemical propulsion), it can generate thrust in a vacuum.

For a rocket engine carrying its own propellants, the thrust is

F_T = ṁv_e + (p_e − p_a)A_e

Symbol	Physical meaning	SI unit
FT	Engine thrust	N
ṁ	Propellant mass flow rate	kg/s
ve	Exhaust velocity relative to engine	m/s
pe, pa	Exit and ambient pressure	Pa
Ae	Nozzle exit area	m²

The first term represents momentum thrust; the second is pressure thrust. A nozzle converts thermal energy into directed exhaust kinetic energy. Its geometry influences exhaust velocity and pressure, and therefore engine performance.

Worked example. For ṁ = 250 kg/s and v_e = 3,000 m/s, assuming zero pressure contribution, F_T = (250)(3,000) = 750,000 N = 750 kN.

02 / ASCENTForces acting on a rising rocket

During vertical powered ascent, the major forces are thrust upwards, gravity downwards, and aerodynamic drag opposite the velocity. Newton's second law gives a simplified instantaneous model:

ma = F_T − mg − D

A vertical launch requires thrust to exceed weight at the moment of liftoff:

TWR = F_Tmg > 1

For a 50,000 kg rocket producing 750 kN, its initial thrust-to-weight ratio is approximately 1.53, and its initial acceleration without drag is 5.19 m/s². As propellant is consumed, the mass decreases, so acceleration can rise even if thrust stays constant.

03 / VARIABLE-MASS MECHANICSThe Tsiolkovsky rocket equation

The central challenge of rocket flight is that the rocket becomes lighter while it burns propellant. Conservation of momentum yields, for an ideal rocket with constant effective exhaust speed and no external forces:

m dv = −v_eq dm

Separate variables and integrate from initial mass m₀ to final mass m_f:

∫ dv = −v_eq ∫ (dm / m)

Δv = v_eq ln(m₀ / m_f)

Since the effective exhaust speed is v_eq = I_spg₀, with g₀ = 9.80665 m/s²:

Δv = I_spg₀ ln(MR)

Here MR = m₀/m_f is the mass ratio, and I_sp is the specific impulse in seconds. Because the equation is logarithmic, much larger propellant fractions are necessary to achieve additional velocity increments.

Model: ideal delta-v

Figure 1. Ideal velocity increment versus initial-to-final mass ratio. Curves are calculated directly from the Tsiolkovsky equation; gravity and aerodynamic losses are excluded.

*Assumes the mass lost is solely expelled propellant; final mass includes dry structure and payload.

04 / PROPULSIONSpecific impulse and engine performance

Specific impulse describes how effectively a propulsion system uses its propellant. It is defined using standard gravity (not the local gravitational acceleration):

I_sp = F_Tṁg₀

Higher I_sp generally means more total impulse per unit propellant weight. However, a high-I_sp engine does not necessarily have high thrust: chemical launch engines and electric spacecraft thrusters serve very different missions.

Illustrative Isp	Equivalent exhaust velocity
250 s	2.45 km/s
300 s	2.94 km/s
350 s	3.43 km/s
450 s	4.41 km/s

These are calculated examples, not measured performance data for specific engines.

05 / VEHICLE DESIGNWhy rockets have multiple stages

Carrying empty tanks and spent engines requires extra propellant. In a multistage launcher, exhausted stages are released so the remaining vehicle accelerates with less inert mass. The ideal velocity increments of the individual stages add:

Δv_total = Σ [I_sp,i g₀ ln(m_0,i/m_f,i)]

Figure 2. Simplified conceptual illustration of staging and spacecraft deployment. Not to scale.

Actual mission design also accounts for separation events, structural limits, guidance, and the mass of carried upper stages and payload.

06 / ORBITAL PHYSICSReaching space is not the same as reaching orbit

To remain in a circular orbit, a spacecraft needs sufficient tangential velocity to continuously fall around Earth. Gravity provides the required centripetal acceleration:

GMmr² = mv²r

v_orb = √(GM / r) = √(μ / (R + h))

Here μ = GM is Earth's standard gravitational parameter, R is the mean Earth radius, and h is altitude above that idealized spherical Earth.

Model: orbital speed and altitude

Figure 3. Ideal circular-orbit speed against altitude, using μ = 398,600.4418 km³/s² and R = 6,371 km. Atmospheric drag, Earth's oblateness, and orbital perturbations are neglected.

At 400 km altitude, the ideal circular-orbit speed is about 7.67 km/s, or 27,600 km/h. A launch vehicle needs a larger total delta-v budget because of gravity losses, aerodynamic drag, steering and mission constraints.

07 / ENERGYEscape velocity

Escape speed comes from conservation of mechanical energy. At the minimum escape condition, an unpowered object reaches an infinite distance from Earth with zero remaining speed:

12mv² − GMmR = 0

v_esc = √(2GM / R)

At the Earth's surface, the result is approximately 11.2 km/s. This is not a required instantaneous liftoff speed: a powered spacecraft can gain energy over time through continuous or repeated engine burns.

08 / ATMOSPHERIC FLIGHTDrag, dynamic pressure and Max-Q

Atmospheric drag depends on air density, the square of air-relative speed, reference area, and drag coefficient:

D = ½ρv²C_DA

Dynamic pressure is q = ½ρv². Early in ascent, speed increases while atmospheric density decreases with altitude. Their competing effects often produce a maximum dynamic pressure known as Max-Q. Rockets may throttle their engines during high-load portions of the climb to protect the vehicle.

09 / GUIDANCEThe gravity turn: from vertical flight to orbit

Rockets initially climb close to vertically to clear the launch facility and dense lower atmosphere. They then tilt and build the horizontal speed required for orbit. A carefully planned gravity turn can reduce the steering effort needed during much of the ascent.

01Liftoff

02Pitch over

03Max-Q

04Stage separation

05Orbit insertion

Exact timing and trajectory depend on rocket design, launch site, payload and target orbit; this sequence is conceptual rather than a universal flight schedule.

10 / TECHNOLOGYChemical versus electric propulsion

Chemical rockets

Propellants release chemical energy and produce large exhaust mass flow rates, enabling high thrust. They dominate launches from planetary surfaces and are also used for major orbital manoeuvres.

Electric propulsion

Ion and Hall-effect thrusters use electrical energy to accelerate charged particles. They commonly deliver low thrust but much higher specific impulse, making them valuable for long-duration in-space missions.

11 / DATAKey Earth and orbit reference values

Quantity	Approximate value	Context
Mean Earth radius	6,371 km	Spherical model
Earth gravitational parameter	398,600 km³/s²	Standard value, rounded
Standard gravity	9.80665 m/s²	Conventional reference
Circular speed at 400 km	7.67 km/s	Ideal two-body orbit
Orbital period at 400 km	92.4 min	Ideal two-body orbit
Escape speed at surface	11.2 km/s	No atmospheric resistance

12 / CONCLUSIONPhysics makes space exploration possible

Rocket engineering unites principles that students encounter throughout physics: momentum conservation explains thrust, variable-mass dynamics determines available delta-v, thermodynamics and fluid dynamics govern the engine, and gravitation sets the conditions for orbital flight and escape.

Rocket science is not magic: it is physics in action. Every successful mission is a practical demonstration of those laws — and of the engineering required to use them.

REFERENCESSources and further reading

1. NASA Glenn Research Center. (n.d.). Rocket thrust equation. NASA Glenn.
2. NASA Glenn Research Center. (n.d.). Ideal rocket equation. NASA Glenn.
3. NASA Glenn Research Center. (n.d.). Specific impulse. NASA Glenn.
4. NASA Glenn Research Center. (n.d.). Guide to rockets. NASA Glenn.
5. Sutton, G. P., & Biblarz, O. (2017). Rocket propulsion elements (9th ed.). Wiley.
6. Curtis, H. D. (2020). Orbital mechanics for engineering students (4th ed.). Butterworth-Heinemann.

EUFISICA · José Gonçalves
Physics · Science education · Space exploration

Houston we have a problem... in Mars

This is a theoretical problem on a Mars habitat and the astronauts have to move it from the planet.
Check the full reading:

Physics of Superheroes

In the past, I posted about the Physics of Spiderman 3, Superman. 

Today, I present in this blog a paper about Physics and Superheroes.

The Standard Model

Standard Model - Source: Wikipedia

 The Standard Model of particle physics is the theoretical framework that describes the fundamental building blocks of the universe and the forces through which they interact. It is one of the most successful and rigorously tested theories in modern physics.

Here is a breakdown of how the universe is put together according to the Standard Model.

1. The Building Blocks: Matter Particles (Fermions)

All tangible matter in the universe is made up of particles called fermions, which have half-integer spin (e.g., 1/2). These are divided into two main categories, each split into three "generations" of increasing mass.

Quarks

Quarks are heavy particles that experience the strong nuclear force. They combine to form composite particles called hadrons (like protons and neutrons).

• Generation 1: Up (u), Down (d) — These make up ordinary stable matter.

• Generation 2: Charm (c), Strange (s)

• Generation 3: Top (t), Bottom (b)

Leptons

Leptons do not feel the strong nuclear force.

• Generation 1: Electron (e^-), Electron Neutrino (ν_e)

• Generation 2: Muon (μ), Muon Neutrino (ν_μ)

• Generation 3: Tau (τ), Tau Neutrino (ν_τ)

Note: For every matter particle, there is a corresponding antimatter particle with the same mass but opposite electric charge (e.g., the positron is the antielectron).

2. The Messengers: Force Carriers (Gauge Bosons)

Particles interact by exchanging force-carrier particles called gauge bosons, which have integer spin (e.g., 1). The Standard Model accounts for three of the four fundamental forces of nature:

Force	Description	Gauge Boson (Carrier)	Mass / Range
Electromagnetism	Governs atomic structure, light, and chemical reactions.	Photon (γ)	Massless / Infinite
Strong Nuclear Force	Binds quarks inside protons/neutrons and holds atomic nuclei together.	Gluon ($g$)	Massless / Short-range
Weak Nuclear Force	Responsible for radioactive decay (like beta decay) and initiating solar fusion.	W+ / W- and Z0 Bosons	Very heavy / Ultra short-range

3. The Mass Giver: The Higgs Boson

The Higgs Boson (H) is a scalar boson (spin 0) associated with the Higgs field, which permeates the entire universe.

• Mechanism: As fundamental particles move through this field, they interact with it. The strength of this interaction determines the particle's inertial mass.

• Particles like the top quark interact strongly and are very heavy; photons do not interact with it at all and remain massless.

• Its discovery at CERN's Large Hadron Collider (LHC) in 2012 was the final piece confirming the Standard Model's framework.

What the Standard Model Left Out

While incredibly robust, the Standard Model is known to be an incomplete theory of nature because it fails to explain a few massive cosmological puzzles:

• Gravity: It does not include general relativity. The hypothetical carrier of gravity, the graviton, has not been incorporated mathematically into the quantum framework.

• Dark Matter & Dark Energy: The model only accounts for about 5% of the energy-mass composition of the observable universe. The remaining 95% is completely unaccounted for.

• Neutrino Mass: In the original mathematical formulation of the Standard Model, neutrinos are massless. However, oscillation experiments have proved they possess tiny, non-zero masses.

• Matter-Antimatter Asymmetry: It doesn't fully explain why the universe today is overwhelmingly made of matter, even though the Big Bang should have produced equal amounts of matter and antimatter.

Physics and Cristiano Ronaldo

 

Jupiter Notebook File (PT)

Eggnaut challenge

 

Eggnaut protection created with Gemini

One of the students’ favorite challenges is the Eggnaut experiment. In this activity, they use the physics learned in secondary school along with engineering, management, economics, and other skills to create a cost-effective solution to protect the egg.

You can check the "Jupiter" book in python here: Simulations/eggnaut.ipynb at main · eufisica/Simulations