*The Principle of the Rocket*

Most people have at least an intuitive notion of the center of gravity (CG) of an object: it is the point on which the object can be perfectly balanced. Grab a broom at one end and the other end tries to drop down; grab it at its center of gravity, and it stays balanced, neither end tipping over. |

By Mach's formulation of the equations of motion, if the heavy ball receives an acceleration

*a*, then the light one gets*2a*, twice as much. For each increment in the velocity of the heavy ball, the light one receives twice as much, and it follows that at any time, its total velocity,**as well as the distance covered**, are twice those of the heavy ball. Conservation of momentum leads to the same result. If then the heavy ball is at a distance D from the initial position of the spring, the light one is at distance 2D--as in the earlier

**figure**, reproduced here. No matter how much time passes,**the center of gravity stays at the same spot**.*Rockets*

Launch of an Atlas-Centaur. |

**Note:**Most of the preceding came from an early version of "

**Stargazers**", which tried to avoid explicit use of the

**conservation of momentum**. This was later amended by a section

**Smoment.htm**on the conservation of momentum, and an associated

**Lesson plan**

**Lmoment.htm**. The section below was copied from Lmoment.htm, where it appears as an optional addition.

*Rocket Motion*

Suppose we have a rocket of total mass

**2M**, of which**M**is payload and**M**is fuel.As the fuel is burned, it is ejected with some constant velocitywrelative to the rocket, creating (we assume) constant thrust. Let us simplify matters by also assuming the launch is from some point in space, so that the thrust of the engine only has to overcome the rocket's inertia. In launches from the ground, part of the thrust is needed to overcome gravity too--see Section #18.

The rocket accelerates gradually. Starting from rest, its moves rather slowly at first. After a while, however, not only is its velocity greater, but its acceleration has grown too: at first, nearly the entire mass2Mmust be accelerated, but as fuel is used up, the mass being accelerated is less and less. By the time a massMof fuel has been burned--half the starting mass of the rocket--its acceleration has doubled, because the same push is applied only to a mass M. Its velocityVat this point may be significant, but to calculate it (givenwand the rocket's thrust) requires calculus, so let us just assume we have somehow got that value.

The payload now has gained velocity

**V**. But we need

**more!**So we build a rocket of mass**4M**, of which**2M**is fuel, while the payload, also of mass**2M**, is the smaller rocket described above, serving as second stage, with half of its mass also given to fuel (to simplify the calculation, we neglect the mass of the bigger rocket itself, although it, too, needs to be accelerated). When the fuel of the big rocket is finished, we reach a velocity**V**, then the second stage is ignited, adding another**V**to the velocity of the payload, for a total of**2V**.**Still faster!**Now the rocket has mass**8M**of which**4M**is fuel of the first stage, while**4M**is the two-stage rocket of the preceding design. The first stage gives velocity**V**, to which the other two add**2V**, for a total of**3V**. By now you can see the trend. If the mass of the final payload is M, then

Total mass | Gives final velocity |

2M | V |

4M | 2V |

8M | 3V |

16M | 4V |

32M | 5V |

64M | 6V |

Each time the velocity increases by one notch, the mass

Staging of the rocket makes the process more efficient. If the rocket of total mass

However, the initial rocket engine for accelerating

**doubles**.Staging of the rocket makes the process more efficient. If the rocket of total mass

**8M**(say) had just one engine and it burned**7M**of its fuel, it seems as if the same effect would be obtained as firing 3 identical rockets simultaneously. (We still need divide the calculation into stages--say, the burning of the first**4M**, then the burning of**2M**, then of the remaining**M**. Each time, less mass needs to be accelerated).However, the initial rocket engine for accelerating

**8M**would have to be very big, and that creates at least two problems. First, it would be heavy, and towards the end of the burn, it is inefficient to carry along such a massive engine (and a big empty fuel tank, too), in addition to the payload. And second, because it is powerful enough to lift**8M**(and rockets cannot very well adjust their thrust), at the end it creates a large acceleration, subjecting the structure of the rocket to a much greater force (see example in Section #18). It is better therefore to drop the big empty tanks and large engines along the way, and continue with smaller ones.(By the way, the "Atlas" rocket pictured above kept its very lightweight stainless steel tank all the way, but dropped two of its three engines).But the basic pattern remains: the final velocity grows much more slowly than the mass of the rockets required by it (like the logarithm of the mass, if this means something to you). A rigorous derivation (which uses calculus) gives the equivalent of a huge number of little stages, fired one after the other, but leads to the same conclusion.

**This is one of the great problems of spaceflight**, especially with the first stages which rise from the ground:

**even a small payload requires a huge rocket**. Perhaps some day space explorers will be able to shave off some fuel weight by using air-breathing rockets ("scramjets") but those seem practical only for the lowest 1/4 to 1/3 of the orbital velocity. Launching from a high-flying airplane--like Burt Rutan's "SpaceshipOne" (using a clever combination of liquid and solid fuel), or the "Pegasus" solid-fuel rocket, used in launching some small satellites--also helps cut air resistance, another factor. But no other shortcuts are in sight. Once in orbit, of course, more efficient but more gradual ways of generating thrust can be enlisted, like

**ion propulsion**.

**As noted above**, actual rockets are staged more carefully, with each stage designed separately. We do not just place one rocket on top of two, which go on top of 4, which go on top of 8, etc.

**However**when the first US satellites were launched, there was no time for sophistication: Russia had successfully beaten the USA to the first launch, while the carefully designed "Vanguard 1" crashed and burned during launch, in full view of TV. Von Braun, leader of the US military rocket program, quickly assembled a 4-stage launcher to fill-in for "Vanguard.". The

**first**stage was a modified big "Redstone" missile, the

**second**stage, a cylindrical cluster of 11 solid-fuel "Sergeant" rockets strapped together, the

**third**stage 3 "Sergeants" strapped together (in the middle of the cylindrical cluster) and the

**fourth**stage a single "Sergeant" which stayed attached to the payload. It worked, and the rest is history.

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