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# Advanced Math/Model Rocket Altitude Equation

Question
At the page http://www.rocketmime.com/rockets/rckt_eqn.html there is a calculation that has a variable I don't understand. This variable is called "In" and there is nothing on the page to reference it, and I have never seen something like it before. I know it's value to be about 1.784 I believe, but no clue how they get it. I think maybe it's an Iteration? Anyway, would you be able to explain this to me? The formula and values are below.

yb = [-M / (2*k)]*ln([T - M*g - k*v^2] / [T - M*g])
= [-0.05398 / (2*0.000217)]*ln([6 - 0.05398*9.8 - 0.000217*118.0^2] / [6 - 0.05398*9.8]) = 99.95 m

Hi, The variable you are curious about is actually a function, namely the natural logarithm. It is written as "ln", or "el en" (for "log natural", I suppose).

The log function tells you the exponent that you need to raise a number (the base) to get the original number. I know, way too confusing. But you have seen this type of thing for base 10; if I write 100 = 10^2, then 2 is the exponent I need to raise 10 to get 100. So if I take the log to the base 10 of 100, the answer is 2

log(100) = 2.   <--  note that I have written "log" for the base 10 case; it is "ln" for the natural log case.

The natural log has the base e ≈ 2.718281828459045... (not to the base 10). This is a number that comes up naturally in mathematics and is closely related to the exponential function, exp(x). In this case

let y = exp(3) = e^3 = 20.08553692318766..., so that ln(20.08553692318766) = 3.

For the equation you are interested in, we have

ln([T - M*g - k*v^2] / [T - M*g])

ln([6 - 0.05398*9.8 - 0.000217*118.0^2] / [6 - 0.05398*9.8]) = ln(4.438) = 1.490

or

e^(1.490) = 4.4370.

Hope this helps.

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#### randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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