Tsiolkovsky Rocket Equation — the Math of Reaching Orbit

1. Derivation from Newton's Third Law

A rocket accelerates by ejecting mass backward. At instant t, the rocket has mass m and velocity v. It ejects propellant at exhaust velocity v_e relative to the rocket:

2. Specific Impulse

Specific impulse I_sp is the thrust produced per unit weight of propellant consumed per second — the "fuel efficiency" of a rocket engine:

3. Mass Ratio and the Exponential Problem

To achieve orbital velocity (~9.4 km/s with losses), rearranging gives the required mass ratio:

4. Staging: Beating the Exponential

By discarding empty propellant tanks and engines (structural mass), staging multiplies the effective mass ratio:

5. Gravity and Drag Losses

6. Propellant Choices

7. JavaScript Rocket Simulator

// Tsiolkovsky rocket simulator with gravity + drag
function simulateRocket({
  m_dry,       // dry mass [kg]
  m_prop,      // propellant mass [kg]
  Isp,         // specific impulse [s]
  thrust,      // thrust [N]
  dt = 0.5,    // timestep [s]
  Cd = 0.3,    // drag coefficient
  A = 10       // cross-section area [m²]
}) {
  const g0 = 9.80665, R_E = 6.371e6;
  const ve = Isp * g0;
  const mdot = thrust / ve; // mass flow rate [kg/s]
  let m = m_dry + m_prop;
  let v = 0, alt = 0;
  let t = 0;
  const log = [];

  while (m > m_dry + 0.1 && alt >= 0) {
    const g = g0 * (R_E / (R_E + alt)) ** 2;
    const rho = 1.225 * Math.exp(-alt / 8500); // exponential atmosphere
    const drag = 0.5 * rho * Cd * A * v * Math.abs(v);
    const F_net = thrust - m * g - drag;
    const a = F_net / m;
    v += a * dt;
    alt += v * dt;
    m -= mdot * dt;
    t += dt;
    log.push({t, alt, v, m});
  }
  return log;
}

// Falcon 9 first stage approximation
const flight = simulateRocket({
  m_dry:   25 600,    // kg
  m_prop: 395 700,    // kg RP-1 + LOX
  Isp:         282,    // sea-level average
  thrust:  7_607_000, // N (9 Merlin engines)
  dt: 0.5
});
const meco = flight[flight.length - 1];
console.log(`MECO: alt=${(meco.alt/1000).toFixed(1)} km, v=${meco.v.toFixed(0)} m/s`);

// Ideal (vacuum, no gravity):  Δv = Isp·g₀·ln(m₀/m_f)
const deltaV_ideal = 311 * 9.80665 * Math.log((25600+395700) / 25600);
console.log(`Ideal Δv = ${(deltaV_ideal/1000).toFixed(2)} km/s`); // ~9.4 km/s

8. Real Rockets and Mission Design

• Gravity assist: Swing-by a planet to gain velocity "for free" (exchanged from the planet's orbital energy). Voyager 2 used Jupiter, Saturn, Uranus to reach Neptune — this would be impossible with propellant alone.
• Delta-v budget: Every mission is planned as a sequence of Δv manoeuvres: launch, TLI, LOI, landing. Total Δv determines propellant mass via Tsiolkovsky.
• Reusability: SpaceX Falcon 9 reserves ~400 m/s Δv for boostback and landing burns — reducing payload but amortising vehicle cost over many flights.
• ISRU (in-situ resource utilisation): Producing methane/oxygen on Mars from CO₂ and water ice eliminates the return-trip propellant from the mass ratio, enabling human return missions.
• Electric propulsion: Ion thrusters (Isp ~3000 s) dramatically reduce propellant mass but provide millinewtons of thrust — practical only for deep-space probes or station-keeping.