Interactive Rocketry Tutorial

THE ROCKET EQUATION

The rocket equation is the main equation used in rocketry. It was derived by a Russian scientist called Konstantin Tsiolkovsky in 1903.

On the Earth, the rocket equation is quite complicated because you have to consider the aerodynamic drag of the rocket (due to the atmosphere) and Gravity. In space, we can forget about drag (there is no air in space).

Consider then, a rocket in space. Before the rocket fires its engines, it has an initial mass, minitial. As the rocket is firing its engines, moving upwards against gravity, g, it has a specific impulse, Isp. After it has fired its engines, and used up the chemicals which are its fuels, it has a final mass, mfinal.

The importance of this is the Isp and the value for minitial divided by mfinal (called the Mass fraction). The larger the mass fraction, the better, but this needs the rocket to be very lightweight, and mostly fuel.

The easy way to achieve a large mass fraction is to use multi-stage rockets, but as mentioned earlier, these are not the best or most cost effective way to get into space (a bit like using a Jumbo Jet once, and then throwing it away).

The best way is to use reusable rockets, and these need to be more efficient i.e. have a higher specific impulse. With all these values above, we can calculate the change of velocity of the rocket, using the rocket equation:

where dV is the change in velocity, and ln is the Natural Logarithm.

The value of dV is 11 kilometres per second for a rocket to get into orbit ! (including the effects of gravity and air resistance).

QUESTION

A crew of astronauts in a spacecraft are about to fire their rocket engines, to catch up with a Near Earth asteroid the size of a small city. Their rocket engines have a specific impulse of 450 seconds, the acceleration due to gravity is 10 metres per second squared, and the initial spacecraft mass is 100,000 kg, and the final spacecraft mass is 50,000 kg (after the engines have been fired). Using the rocket equation, find out what the change in velocity is ?

This page is maintained by Richard Osborne