Usually when on Earth we can use as an approximation:  g = a = 9.18 $\frac{m}{s^2}$ s = vi + 1/2 a$t^{2}$  However, how would I determine the distance traveled in situations where the the acceleration due to gravity is changing non-negligibly because the distances are so large? I know that the following equation has to be used somehow:  a = $Gmr^{-2}$  I am not sure exactly how though. For example, if given the information that an object is at a certain distance and is being attracted by a larger object of a given mass, the method should calculate where the object will be at a given time.
I thought this would be a simple question, but I'm having trouble figuring it out. Not a homework assignment btw. I am a physics student and am just genuinely interested in physics problems involving math, which would be all of them.  So lets say we drop an object from a height, $R+r$, it falls toward earth. This height, $R+r$, is far enough away that the $g$ it experiences is a fraction of $g$ at sea-level. Let's just say that air resistance is negligible, and it wouldn't be that complicated to just integrate from $0$ velocity to the terminal velocity piece wise and deal with the rest of it later.  So the key here is that acceleration is changing with time. I thought I could simplify this by saying it changes with distance, and it has nothing to do with time, but this didn't really help, my guess is that maybe time is important (doh).  I tried integrating acceleration with time, and ended up nowhere. I tried integrating $a=GM/R^2$ with respect to $R$ from $R+r$ to $R$ and ended up with a negative function.  I saw somewhere someone tried to expand with taylor series, they even have something similar on hyperphysics, but I can't figure out how to obtain the polynomials that precede the variables.    This is the hyperphysics site where they use polynomials to find the distance.   Maybe I can't solve this because I haven't taken a course in differential equations yet. What I want to know is how to calculate the distance at any time.
I read a collection of Sci-Fi short stories back in the 1970's or early 80's that had one about a couple in love who kept losing each other. The book was my dad's so the story itself could be much older than that. I am very fuzzy on the details as I was young when I read it and it was so long ago. I believe the woman/girlfriend was "taken" (by some bad guy aliens??) while she was sleeping. I think it involved time travel and the man had to search through time to find her. After he found her, when he was asleep one time, he somehow was then "taken" and I think transported through time and had to fight his way back to her again. He may have been some sort of police officer. I just remember in the end, at night when they went to bed, the couple always wore handcuffs so they were never separated when they went to sleep ever again. Does this ring a bell with anyone?  (There is a possibility it is in a short story collection that also contained " The Twonky " by Lewis Padgett (1942), as I remember reading that story around the same time) How to define servlet filter order of execution using annotations in WAR Finding the sum of series $\sum_{n=0}^∞ \frac{2^n + 3^n}{6^n}$ I would like to use Flash Player to show web contents but since today (12 Jan 2021) Adobe is blocking it. Is there a way to use it? I've tried Firefox, Chrome, Safari and Brave browsers. When applying to a PhD program in the US, how does the admissions process work? If an applicant is weak in a particular area, is it possible to offset that by being strong in a different area?  Note that this question originated from this . Please feel free to edit the question to improve it. What does it do exactly if I click in the window of cmd? Suppose I run cmd.exe and ping localhost. After that I ping localhost again but I click into the window and the ping stops until I press enter.  What happens exactly when I click into the window? Does it suspend the process or something else? I attached a screenshot to clarify this issue.  Note: I do not actually click but drag using the mouse which creates a white box in the window as you can see on the screenshot.  Edit: further clarification: What I really wish to know is whether dragging the window halts the cmd process or not?  Edit: even more clarification: I know that I'm using Quick Edit and pressing enter copies the contents I selected. What I do not know is what happens in the background. Mapping of zero vector under a linear transformation Let $T$ be a linear transformation from a vector space $V$ to $W$ that are defined over a field $\mathbb F$.   Now, $T(\mathbf0)=\mathbf0$.   Which implies the zero vector in $V$ can't move or be transformed to a new vector in $W$. Why is it so?   Does it also have a geometrical significance? n tasks assigned to n computers, what is the EX value of a computer getting 5 or more tasks? Difference between "irascible", "fractious", "irritable" and "atrabilious"? It seems that they can all mean "easily provoked to anger"  irascible：Easily provoked to outbursts of anger; irritable.  fractious：Irritable; argumentative; quarrelsome.  irritable：1.Capable of being irritated. 2.Easily exasperated or excited.  atrabilious：1.Characterized by melancholy. 2.ill-natured; malevolent. Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: $$f(z+\omega_1)=f(z)=f(z+\omega_2), \quad \forall\,z\in \Bbb C.$$Prove that $f$ is constant.   The intuition seems clear to me, we have the three vertices of a triangle given by $0$, $\omega_1$ and $\omega_2$. All points in the plane are one of the vertices of that triangle under a suitable parallel translation. The constant value will be $f(0)$, fine. Throwing values for $z$ there, I have found that $$f(n\omega_1) = f(\omega_1) = f(0)=f(\omega_2) = f(n\omega_2), \quad \forall\, n \in \Bbb Z.$$  I don't know how to improve the above for, say, rationals (at least). Some another ideas would be:   Checking that $f' \equiv 0$. I don't have a clue of how to do that. Write $w = a\omega_1+b\omega_2$, with $a,b \in \Bbb R$, do stuff and conclude that $f(w) = f(0)$. This approach doesn't seem good, because I only have a weak result with integers above. Finding that $f$ coincides with $f(0)$ on a set with an accumulation point. This seems also bad: the set on with $f$ coincides with $f(0)$ by which I found above is discrete.    Nothing works and this is getting annoying... And I don't see how analyticity comes in there. I'll be very thankful if someone can give me an idea.  (On a side note.. I know that this title is not informative at all. Feel free to edit if you come up with something better.) Why isn't .profile sourced when opening a terminal? Just installed Ubuntu 14.04.1 LTS.  According to .profile,  # ~/.profile: executed by the command interpreter for login shells.  # This file is not read by bash, if ~/.bash_profile or ~/.bash_login exists.   There is no ~/.bash_profile or ~/.bash_login.  It does not seem to source .profile when opening a terminal.   Why doesn't it source .profile when opening a terminal? Why is .profile there if it is not sourced? How do I know/control what is sourced when a new terminal is opened? When a new terminal is opened, does it do a "login"? Where is a good place to add $HOME/bin to my path? .bashrc? Will that cause a problem because .profile has the same code? 408 Request Time-outs still continues I'm frequently getting 408 Request Time-out errors  when browsing through Stack Overflow and Meta. I used to get this four or five times a day before, but now it has become very frequent. I've read  in which Kyle Brandt ♦ says:     We send our web logs to SQL so we can query them. It turns out that we have a bug that prevents 408s from being logged to SQL. [...] I will do more extensive research when our 408s are part of queryable logs (should be a week or two [...]).   My internet connection is somewhat stable and I haven't experienced this issue on other sites before. I don't think it's a browser-specific issue as it is reproducible on latest versions of Google Chrome and Firefox (on Ubuntu 13.10). However, it is specific to Stack Overflow and Meta. I'm unable to reproduce this on other SE sites.  Am I the only one facing this issue? Could it be something wrong with the regional CDN or something like that? Could someone please take a look at this issue? ()  This appears when I try to close-vote questions, too: Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$ Network simplex method, leaving and entering variables Man undergoes radical surgery to research aliens How to politely negotiate PhD funding offer? I have been fortunate enough to get accepted with 5 years guaranteed funding to my 2nd choice graduate school. While the guarantee is great, the funding is only for 9/months out of the year. Would it be rude to ask them to guarantee summer funding to me for the first year until I can write a grant to get on a RAship? I know I should be more appreciative, since not everyone gets funding, but just an increase of about ~3k a year would make living there so much better and I do not want to accept an offer at a lesser institution just because of money.  What would be the most polite way of doing this?  I want to make it clear that I am not trying to get more money for the sake of getting more money, moreover that I am just not comfortable with so little money and such a high living cost in that area.  Thank you for your time. According to some source or other (I forget which now) it is theoretically impossible for bumblebees to fly by virtue of their size/bulk/aerodynamic properties. Is this old adage apocraphyal or true? And if its true, how come they do fly? Equal simple field extensions? I have a question about simple field extensions.  For a field $F$, if $[F(a):F]$ is odd, then why is $F(a)=F(a^2)$? Using the code below I add text to end of file names which are:  1.mp3 2.mp3   and should be:  1text.mp3 2text.mp3   but it changed to:  1.mp3text 2.mp3text   Code:  @ECHO ON  FOR %%A IN ("C:\Users\user123\Desktop\mp3\*.mp2") DO (     CALL :RenameFiles "%%~A" "%%~NXA" ) GOTO EOF  :RenameFiles  SET fname=%~2 SET renname=%fname=% REN "%~1" "%renname%text%" GOTO E I have a scenario where i want to calculate sum of column   SAL|CAL|TAG|VAL 12.11"|"1.1"|"2.1"|"1.1 13.11"|"1.1"|"2.2"|"2.2 14.11"|"1.1"|"2.4"|"2.4   My below code not giving output giving output as 0.00  I need output for this column CAL VAL  awk -F'|' '{T+=$2} END { printf "%.2f\n", T }' demo.txt   output :  6.7 5.7