Sciency Words: Ideal Rocket Equation

Sciency Words MATH

Today’s post is part of a special series here on Planet Pailly called Sciency Words. Every Friday, we take a look at a new and interesting scientific term to help us all expand our scientific vocabularies together. Today’s word is:


April is Earth month here on Planet Pailly, but after two weeks of blogging about the planet Earth, I’m ready to move on.

Ap08 Earth's Pull

Unfortunately, escaping Earth’s gravity is far easier said than done. The high, high cost of getting to space can be quantified using something called the ideal rocket equation (also known as Tsiolkovsky’s rocket equation or simply the rocket equation).

The equation is as follows:

∆v = vln(m0/m1)

Delta-v (∆v) represents the total change in velocity you’re aiming to achieve in any rocket-propelled maneuver, including liftoff. In order to reach low Earth orbit from the ground, your delta-v must equal at least 9.4 kilometers per second. To get that value, you’ll need to adjust the other variables in the equation.

  • Initial mass (m0): The total mass of your spacecraft plus the mass of your fuel and fuel tanks.
  • Final mass (m1): The total mass of your rocket after the maneuver is complete.
  • Effective exhaust velocity (ve): This is basically how much thrust your rocket can produce.

Increasing your rocket’s initial mass (by adding more fuel) will help increase your delta-v. Decreasing your final mass (by not only using up fuel but also shedding empty fuel tanks as you go) will also increase your delta-v. In fact, the greater the difference between the initial and final mass, the greater your delta-v will be, according to this equation.

However, increasing the difference between initial and final mass only creates a logarithmic increase in delta-v (the “ln” part of the equation is a natural logarithm). This means that adding more and more fuel produces diminishing returns. At some point, this is no longer a cost effective way to increase your delta-v.

Your other option is to use a more energetic fuel, increasing your effective exhaust (ve). Unfortunately, modern rockets already use some of the most effective chemical fuels available. With current technologies, the only way to significantly improve the ve part of the equation is with nuclear powered rockets, which might raise a few safety concerns, to say the least.

What Does All That Mean?

Due to the rocket equation, fuel constitutes 80 to 95% of a rocket’s mass at launch. Even a tiny satellite requires absurd amounts of fuel to reach space.  This means launching anything into space is expensive (sometimes prohibitively expensive).

The problems associated with the ideal rocket equation are usually glossed over or ignored in science fiction by invoking new technologies or new laws of physics. But embracing the rocket equation and world-building within its limitations could lead to an intriguing setting for a Sci-Fi story. More on that in next week’s edition of Sciency Words.

P.S.: It’s possible that somewhere in the universe, life has evolved on a planet with even higher surface gravity than Earth’s. If so, these aliens would have an even harder time reaching space than we do. In fact, for some alien civilization out there somewhere, the rocket equation may make it effectively impossible to leave their home planet at all.


The Tyranny of the Rocket Equation by NASA astronaut Don Pettit.

Rocket Golf from What If?

7 thoughts on “Sciency Words: Ideal Rocket Equation

  1. Thank Jupiter people are running planet earth, you just never know when a computer is going to flip out on you. We’re not as fast or precise because they remember stuff humans forget but earthers will always have the final say. Thank heavens.


  2. Umm…You do know ion engines in real life escape this, not being normal combustive chemicals. And most of sci-fi switched to ion engines decades ago. And…Also, a lot of these sci fi stories like Star Wars get away with it the same way scientists know solar sails, air-to-space launches with spaceplanes (real technology), and anything that doesn’t use the main thrust of a ship doesn’t count. For Star Wars, the repulsors are a permanently maintained collected of subatomic spacetime knots, which create a repulsive effect in spacetime itself, and shares some reality with spacetime theories which are many.

    And…Most ships only burn fuel when they maneuver or fly the final push to where they can use spacetime manipulation to reach FTL without breaking relativity. Don’t listen to some of the anti-fiction stuff science lovers say. It is rather hypocritical and most people actually don’t know or don’t pay attention to what happens in these kinds of stories.

    Liked by 1 person

    1. I’m not exactly happy to be called “anti-fiction,” but I think you and I would probably agree more than we disagree. I’ve said this before and I’ll say it again: never let science get in the way of a good story.

      However, I do believe it’s important for science fiction writers to know as much as they can about scientific realities before they try to get around them. In this case, science is telling us that lifting yourself out of a planet’s gravity is really, really hard.

      Many Sci-Fi universes try to imagine that problem away, sometimes referencing the kinds of technology you mentioned. I think that’s great, and I give Star Wars a lot of credit for the way they handle these kinds of things.

      But personally, I think it might be fun to tell a story in which the real life challenges of space travel are in full effect and help fuel the conflict. At least that’s something I hope to do someday in my own writing.


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