Understanding the Delta-V formula.

[Rage097]

So. I've been trying my hand at calculating the Delta-V of my craft rather than using guesswork to get to celestial bodies. I am hindered by one part of the formula: Natural Logorithm.

What is it and how is it found?

[blizzy78]

I think this should be your first visit: http://en.wikipedia.org/wiki/Natural_logarithm

Other than that, if you don't really care what it is, it's just a function on your calculator.

[KvickFlygarn87]

In windows' calculator, choose advanced.

Natural logarithm is LN.

[Blue]

An alternative way to do it is to type it out by doing the Log of the mass ratio in base e: or Log of the mass ratio, divided by Log of e.

(log (m1/m0)) / (log (e)) = LN (m1/m0)

[Geschosskopf]

What is it and how is it found?

Others have explained where to find it so I'll answer the other part.

Natural logarithms are based on the number "e", an irrational, endless decimal like pi. e is called "God's approximation of 3" because it seems like much of the universe was built around it. The number crops up in all sorts of scientific formulae. So, to avoid having to worry about how many significant figures of e's endless string to use, the early boffins who discovered e developed the natural logarithm system. By setting e as the base instead of the usual 10, they avoided this problem, which was more of a big deal back in the day before computers. It's the same strategy as using radians instead of degrees to measure angles, so you don't have to mess with all of pi's digits.

Knowing that the universe itself and everyday things like circles are based on irrational numbers can lead to depressing and scary theological conclusions .

[arq]

e^x = sum_{n=0}^<infinity> (x^n)/n!

so e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4!... = 1 + 1 + 1/2 + 1/6 + 1/24... = 2.71828

Anything more than a 4-function calculator will have this function (and the natural logarithm) built in. For KSP, the logarithm (which is the inverse function of the exponential) is much more important

exp(log(x)) = x

Some calculators use log() for the natural logarithm and some use ln(). If log() is not the natural logarithm, it is the 'common' logarithm (base 10 instead of base e).

[Frederf]

e^x is a special function. One of its defining characteristics is that it is a graph of its own slope which is unique if you exclude the obvious case of 0. It's also the exponent base you end up using when you take compound interest and decrease the steps an infinite number of times to produce a smooth continuous compound function.

In any case if you only have Log-base-10 then you can divide the result by Log10(e) which is 0.43429448190325182765112891891661. Windows calculator has a Log-natural button and the "Inv" key accesses all the inverses of all the functions. Cosine becomes Arcosine, ln becomes e^x, and so on. You can change calculator modes under the file menu, view.

[el_coyoto]

A good way to visualize ln(a) is seeing it as the area defined by the 1/x function between 0 and a :

In other words, a sum of increasingly small quantities. It can be used to model certain types of diminishing returns.

You can "feel" it in the delta v equation : when you double you fuel, your delta v does not increase as much.

Hope this helps!

Edited November 7, 2013 by el_coyoto
tablet derping

[numerobis]

You can "feel" it in the delta v equation : when you double you fuel, your delta v does not increase as much.

Actually it means that every time you double your delta V increases by as many m/s as last time you doubled it. But every time you add a tonne of fuel, you only gain half as many m/s as last time you added a tonne.

(Plus complications due to the mass of the fuel tank itself.)

[Specialist290]

In other words, a sum of increasingly small quantities. It can be used to model certain types of diminishing returns.

You can "feel" it in the delta v equation : when you double you fuel, your delta v does not increase as much.

Hope this helps!

Actually it means that every time you double your delta V increases by as many m/s as last time you doubled it. But every time you add a tonne of fuel, you only gain half as many m/s as last time you added a tonne.

(Plus complications due to the mass of the fuel tank itself.)

And to put the same idea you two are talking about in another way: Since the rocket equation is logarithmic, to double the delta-v of a craft by adding more fuel, you want to square your initial mass ratio. Likewise, you can figure out how much fuel you'll have left after spending half of your craft's delta-v allowance by finding the square root of your mass ratio and plugging that into the rocket equation.

Still another way of looking at it: Each ounce of fuel you spend provides slightly more delta-v than the last one, because as your craft burns fuel, it's getting lighter, and thus your engines can provide more acceleration for the same amount of fuel burned.

All that is, essentially, what the "ln" in the rocket equation means in practical terms.

Hope this helps