Skip main navigation

Rocket equation

Watch Jasmina describe how space travel is determined by the rocket equation.
JASMINA LAZENDIC-GALLOWAY: Is it possible to send humans to Mars, even today or in the near future? Well, in theory, yes, but practically, there some difficulties we have to overcome. We have something called the rocket equation that allows us to relate the velocity at which we want our rocket to move, the type of fuel we use for the rocket, and also the mass of the rocket. So we say the final velocity minus the initial velocity is equal to J, which is basically exhaust speed, and then logarithm of your initial mass versus the final mass.
So this is in units of kilometres per second, this is in units kilometres per second, this is in units kilometres per second, and this is in kilograms.
So initial velocity will be 0 because this is where we’re starting from. And let’s say we want to get to 3 kilometres a second. The exhaust speed is dependent on what kind of fuel we use. And with the chemical fuel we use these days, basically, that’s 3. So for the final velocity, we take just the weight of the ship, of our rocket. So let’s say we take 10 to the 6 kilograms, 1,000 tonnes. So then this equation becomes 3 equals 3 logarithm of initial mass over the final mass, which is 10 to the 6. So when we turn this around, we get the initial mass has to be around 2.7 10 to the 6 kilograms.
What this means is basically that we have to have almost three times more initial mass, which will be just in fuel. So in respect to the weight of the rocket, three times that has to be fuel. This is how much it takes just to get to 3 kilometres a second. So as you can see, there are two possibilities to improve our space travel, either to improve J, which is basically moving away from chemical fuels or finding even more efficient chemical fuels, and also to change this ratio of initial mass with the final mass. That’s slightly harder to do because rocket ships have to be quite heavy. So then a lot of effort is put into finding more efficient fuel.
But another way maybe of changing this ratio is not basically changing the type of the rocket, but actually allowing rockets to refuel in orbit.

Watch Jasmina describe how space travel is determined by the rocket equation.

The step-by-step calculations are:

[3,{rm km/s} – 0, {rm km/s} = (3, {rm km/s}) times ln Big( frac{M_i}{10^6, {rm kg}} Big)] [frac{3, {rm km/s}}{3, {rm km/s}} = ln Big( frac{M_i}{10^6, {rm kg}} Big)]

(ln) is the natural logarithm (similar to log), which works in base (e), which is just a number (approximately 2.7). So we use (e) on both sides of the equation to get rid of (ln):

[e^1 = frac{M_i}{10^6, kg}]

and finally rearrange to get (M_i):

[M_i = 2.7 times 10^6, {rm kg}]

Can we overcome this “rocket equation problem” to send humans to Mars? Let’s find out in the next step.

This article is from the free online

How to Survive on Mars: the Science Behind the Human Exploration of Mars

Created by
FutureLearn - Learning For Life

Our purpose is to transform access to education.

We offer a diverse selection of courses from leading universities and cultural institutions from around the world. These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life.

We believe learning should be an enjoyable, social experience, so our courses offer the opportunity to discuss what you’re learning with others as you go, helping you make fresh discoveries and form new ideas.
You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package. Build your knowledge with top universities and organisations.

Learn more about how FutureLearn is transforming access to education