**2. The problem: the Tsiolkovsky equation and its implications**

The basic structure of a rocket is well-known. Demonstrating Newton's Third Law ("For every action, there is an equal and opposite reaction"), when a gas (particularly, a hot, expanding gas) exists within an open-ended chamber, the unequal distribution of forces within the chamber causes the gas to push the rocket forward. **Figure 1** shows the basic application of Newton's Third Law in a "balloon" rocket; chemical rockets work the same way.

Chemical rockets work by burning a fuel (such as liquid oxygen, or a solid combustible compound) inside a thruster. The hot, expanding gases in the thruster push the rocket forward. There is a practical limit to the speeds that can be achieved by chemical rockets, however, because they must "push" all of their own fuel. In 1896, Konstantin Tsiolkovsky, the great Russian rocket scientist, determined the formula for calculating a change in rocket velocity. The formula is:

$$
\Delta \upsilon = \upsilon\_{\epsilon} \bullet \ln \left( m\_{\text{o}} / m\_{f} \right),
\tag{1}
$$

where *v*e is the velocity of the rocket's exhaust, *m*0 is the initial mass of the rocket, and *m*f is the mass of the rocket at the end of the burn [4].

Importantly, because the change in speed of a rocket is a logarithmic function of the ratio of the change in the rocket's mass, and because the change in speed is a direct function of the exhaust's velocity, there is a practical limit to the speeds achievable by chemical rockets. The more mass the rocket must have (in fuel weight), the less it is able to accelerate. For that reason, much of twentieth-century rocket technology involved

**Figure 1.** *Rocket principle (courtesy Mustang Publishing).*

### *Matter/Anti-Matter Propulsion DOI:http://dx.doi.org/10.5772/intechopen.110310*

increasing both the velocity of exhaust and the energy density of fuel, while building multiple stages to achieve escape velocity. (When a stage drops off, the mass of the rocket is decreased, increasing potential acceleration of the rocket for the next given period of time.) The United States' Apollo V rocket, with its dense fuel (liquid oxygen), high-speed fuel pumps, and its multiple stages, is a perfect example of the practical implications of Tsiolkovsky's rocket equation when applied to chemical rockets.

Matter/anti-matter rockets work under the same principle as chemical rockets, only light itself (specifically, 511-keV gamma rays) is the propellant. The velocity of the "exhaust" of a matter/anti-matter rocket is the speed of light; the change in mass of the rocket as it accelerates is virtually zero. For that reason, matter/anti-matter rockets, as a practical implication of Tsiolkovsky's equation, given enough time, can achieve speeds that are a significant fraction of the speed of light.
