Wikipedia:Reference desk/Archives/Mathematics/2020 March 25

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March 25

Integration and physics-related question

Hello, I have a question about integration, which could not be a normal type of math type question since this is an equation related to physics. I was not sure which section I should ask, but I assumed it would be appropriate to ask here. Also, this is my first time using the reference desk, so I am not used to the style of writing an equation.

So, can you decide if my work is correct below?

To note: v(0)=0, x(0)=h, I will integrate from t=0 to t=t dv/dt=-g-bv---(1), v(t)=-(g/b)(1-e^(-bt)) is given(concluded from the former equation (1).) I want to find x(t). Multiply dt to (1) --> dv=-gdt-bdx --> Integrate --> v(t)=-gt-b(x(t)-h) --> x(t)=h-(g/b)(t-(1-e^(-bt))/b)

The answer is correct, but I'm not sure if this logic is correct. v, x is dependant on t, but is this possible? Also, if there are any Wiki pages about this topic(3 or more variables in Differential Equation?)

Thank you very very much, Luke Kern Choi 5 (talk) 15:10, 25 March 2020 (UTC)

First, the writing style here is pretty flexible, as long as you make yourself understood, follow the guidelines listed above and are reasonably polite then I, at least, won't take issue. If you plan on using Wikipedia much in the future then you might want to invest some time into learning some of the math specific markup that's available since it will make reading equations easier. See help:math as a reference for using TeX markup and select 'Math and logic' from drop down menu just below the edit box if you just need a few quick math related symbols.

On the math, I'm assuming x=v' is kind of implied in physics but I didn't see it stated anywhere. Other than that I followed everything except I got different signs for some of the terms in the final expression for x; it could be my mistake but you might want to check it. I believe the solution for v is a case of a first order linear differential equation, see Linear differential equation#First-order equation with variable coefficients for a solution to the general case. It also has constant coefficients so the section Non-homogeneous equation with constant coefficients is also applicable. To get x from v you're basically just applying the Fundamental theorem of calculus and using standard integration techniques. I guess since there are two dependent variables you might want to check Ordinary differential equation#System of ODEs, but it's not really needed in this case because you can solve for one variable at a time. --RDBury (talk) 16:03, 25 March 2020 (UTC)

I got the same signs as in the solution above. It is easier to read if you avoid ${-}\times {-}\times {-}$ by pushing the first minus in, and use a shared denominator $b^{2}$:

$x(t)=h+{\frac {g}{b^{2}}}(1-bt-e^{-bt}).$

Using the Taylor expansion

$e^{-bt}=1-bt+{\frac {1}{2}}b^{2}t^{2}-{\frac {1}{6}}b^{3}t^{3}+O(t^{4}),$

it is now easy to see that

$x(t)=h-{\frac {1}{2}}gt^{2}+{\frac {1}{6}}gbt^{3}+O(t^{4}).$

 --Lambiam 19:13, 25 March 2020 (UTC)

Thank you for answering my first question in the reference desk. For the question, it was about physics(as stated above). Although I got a little overwhelmed(because I am not used to English math words that much, I am high school level), I guess it was a good opportunity to learn! Again, thank you for your great effort, and I'll learn those writing tools. Luke Kern Choi 5 (talk) 23:49, 25 March 2020 (UTC)

FWIW, I found my mistake: reading dv/dt=g-bv, instead of dv/dt=-g-bv. --RDBury (talk) 05:54, 26 March 2020 (UTC)

By going into edit mode, you can see examples of LaTeX-style markup used for typesetting the maths formulas.  --Lambiam 10:43, 26 March 2020 (UTC)