Size a Two-Stage Rocket With the Rocket Equation

The rocket equation is short enough to fit on a napkin, yet it explains why rockets are tall, thin and mostly fuel, and why almost every launcher drops pieces of itself on the way up. This guide sizes an invented two-stage rocket with round numbers, checks it against the Δv needed for low Earth orbit and then tests the design against the usual mistakes. It is a learning exercise, not a design method; real vehicles are sized with trajectory simulations.

The vehicle

The payload is 5 tonnes. The second stage has 4 t of structure and engine and carries 40 t of propellant, burning in vacuum with a specific impulse of 348 s. The first stage has 25 t of structure and engines and 400 t of propellant at an average specific impulse of 300 s over its climb. The total lift-off mass is 5 + 4 + 40 + 25 + 400 = 474 t. All exhaust velocities use g₀ = 9.80665 m/s².

Step 1: the upper stage first

Always start from the top, because each stage has to carry everything above it. The second stage begins with its full propellant, its own structure and the payload: m₀ = 49 t. When it burns out, m_f = 9 t. In the rocket equation calculator, enter 348 s, 49 t and 9 t. The exhaust velocity is 3,412.7 m/s, the mass ratio 5.444 and the natural logarithm of that ratio 1.695, so the stage delivers 5,783 m/s.

Step 2: the first stage

For the first stage the whole loaded second stage and payload count as dead weight. Its wet mass is the full 474 t; at burnout, with its own 400 t gone, 74 t remain. With 300 s the exhaust velocity is 2,942.0 m/s, the mass ratio 6.405 and the logarithm 1.857, so the first stage contributes 5,464 m/s. The ideal total for the vehicle is 5,464 + 5,783 = 11,247 m/s.

Step 3: compare with the mission

Low Earth orbit needs about 7.8 km/s of orbital speed, which the orbital velocity calculator confirms at 7.793 km/s for 185 km. Gravity losses, drag and steering bring the practical requirement to about 9.4 km/s. The design has an ideal 11.2 km/s, a margin of about 1.85 km/s. That margin could be spent on a heavier payload, a higher orbit or reserves, and it is a reminder that rocket equation figures are always optimistic: the losses are real and depend heavily on the thrust of the first stage.

Why staging works

Now imagine the same tanks and engines flown as a single stage that carries all its structure to orbit. The total propellant is still 440 t, but the dry mass at the end is 5 + 4 + 25 = 34 t. The mass ratio becomes 474 ÷ 34 = 13.94 and, at 300 s, the Δv is 7,752 m/s. That is not enough for orbit once losses are counted, even though the vehicle is identical. Staging gained almost 3.5 km/s simply by throwing away 25 t of empty first stage before lighting the second, and by letting the upper stage use an engine optimised for vacuum.

Step 4: sensitivity tests

Two quick checks show how unforgiving the equation is. First, if the first stage is evaluated with a sea-level specific impulse of 282 s instead of the 300 s average, its contribution falls to 5,136 m/s, 328 m/s less. Second, add one tonne to the payload. The second stage now runs from 50 t to 10 t and delivers 5,493 m/s, 291 m/s less, and the first stage, now from 475 t to 75 t, delivers 5,430 m/s, 33 m/s less. The total drops by about 324 m/s to 10,923 m/s. A tonne of payload costs far more Δv on the upper stage than on the lower, which is why upper stages are built to be as light as possible.

Common mistakes

• Forgetting the payload in the upper-stage masses, or forgetting the whole loaded upper stage in the first-stage masses.
• Applying a vacuum Isp to a first stage that spends its first minute in thick air.
• Using log base 10 instead of the natural logarithm, which makes every answer 2.3 times too small.
• Treating the ideal Δv as the orbital speed the rocket reaches, without subtracting gravity and drag losses.

Checklist

1. List the payload and, for each stage, its dry mass, propellant mass and the specific impulse for the conditions it flies in.
2. Work from the top stage down, adding everything above a stage to both its wet and dry masses.
3. Compute each stage’s Δv with the rocket equation and add them.
4. Compare the total with the mission requirement including losses, about 9.4 km/s for low Earth orbit.
5. Test the margin against a lower Isp and a heavier payload before trusting the design.