## The Rocket Equation

To fully appreciate any rocket or rocket-powered spacecraft, a good understanding of one of the fundamental equations of rocket science is essential. The equation relates several basic parameters of spaceflight in one simple formula.

The first item in the formula is the all-important AV, which is the theoretical change in velocity needed for a particular space mission. This value includes not only the velocity required, for example, to reach low Earth orbit, or go to the Moon, but actually incorporates other factors, such as drag and gravity losses, landing maneuvers, and midcourse corrections. For example, a typical AV for getting into low Earth orbit is 30,000 ft/s, which includes about 4,300 ft/s to cover atmospheric drag and gravity losses.

The second factor in the rocket equation is exhaust velocity, c, which is the speed of the exhaust gases as they pass the exit plane of the rocket nozzle. Once past this point, the expanding gases have done all the work they can do in accelerating the rocket. The exhaust velocity depends on the propellant's specific impulse, / , which is the thrust divided by the weight of propellant burned per second. The higher the total thrust per unit weight flow rate of propellant, the higher the specific impulse will be. And the lighter weight the propellant, the higher the exhaust velocity will be. This is why Konstantin Tsiolkovskiy wrote about the spaceflight benefits of hydrogen fuel over a hundred years ago. There is more information on exhaust velocity and specific impulse in the appendix.

Finally, we come to the natural logarithm of the mass ratio, M/m. The mass ratio is simply the initial fully fueled mass M of the vehicle divided by the final mass m, after burning off all propellants. There are two ways to increase this ratio. The first is to maximize the initial weight by using denser propellants, and in fact this technique is used in the first stages of many rockets. The second is to minimize the final weight by making the empty structure of the vehicle as light as possible. One of the greatest challenges in spaceplane design is to get the mass ratio as high as possible in a single vehicle. The easiest method of increasing mass ratio is through a technique known as staging - used by virtually all rockets. The natural logarithm is an exponential curve with a horizontal asymptote, meaning that the higher the mass ratio becomes, the less incremental benefit that accrues. For example, doubling the mass ratio increases the AV, but it does not quite double it, because the logarithmic curve levels out.