# Equation Answers

Okay, here we go!

- This is the Rocket Equation, formulated by the father of rocketry, Konstantin Tsiolkovsky, in the late 1800’s. The equation describes how much of a speed boost a rocket can get — the boost is proportional to the velocity of the gases being shot out of the rocket’s engine, and is also affected by the ratio between the initial fueled weight of the rocket and its final empty weight (including payload). This ratio is why rockets are built with the most lightweight structure possible, and are mostly fuel when they take off. It’s also why many rockets have multiple stages—throwing away an entire piece of the rocket is a great way to reduce weight.
- To fully describe a black hole you need no-holds-barred Einsteinian General Relativity, which involves some very tough math indeed. Fortunately, this simple Newtonion equation can be used to describe the radius of a black hole, also known as the Schwarzschild radius (technically, this equation is only perfectly accurate for non-rotating black holes, but it’s close enough in most cases). The Schwarzschild radius marks the location of the Event Horizon: beyond the horizon not even light can escape the incredible gravitational pull of a black hole. The equation shows that the size of a black hole is directly proportional to its mass.
- The Schrödinger Equation, cornerstone of quantum mechanics. (technically, this is a one-dimensional, time-independent, version of the equation). This compact expression manages to fold in a) the notion that in many environments particles (such as electrons bound in an atom) can only have very particular quantities of energy—this is the “quantum” in quantum physics— b) the idea that particles can be described using waves, and c) the role of chance in quantum theory, as Schrödinger’s equation describes only the probability of finding a particle in a given region of space. To find out exactly where a real particle might be, you have to actually go out and measure it.
- This equation spells out the half-life of a radioactive substance. The half-life is the amount of time that it takes for fifty percent of a given amount of unstable material to decay into something else. Some things have insanely short half-lives—the subatomic particles that are produced in atom smashers like the LHC often have half-lives measured in minute fractions of a second— while others have half lives measured in millenia—the half-life of Uranium-235 nucleii is over 700 million years. Because half-lives are unaffected by chemical changes, they can be used as way to measure time as is done in carbon dating.
- This equation describes an exponential growth curve, the kind of growth pattern that populations of disease-carrying microbes like to follow. Of course, the curve can’t go on forever—sooner or later there’s a crash that isn’t described directly by this formula, as the bug runs out of new hosts to infect.

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this website is wonderful! I finally understand the equation for black holes!