Why do the planets orbit the Sun? If it's because the gravity of Sun then why don't they just fall in and burn up?
The force of gravity is constantly being applied to an orbiting object.  And therefore the object is constantly accelerating.  Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas?  I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it?
This is a repost with additional info of an older question that was never answered. I have stumbled upon this old question while also trying to find the name of the book series myself, as I dearly wish to read it again but for the life of me, I can't remember the name of it.      This is a long shot but I'm really in need of help for this because      it was an amazing book. It was either a kid's or YA book. A guess      from what year it was published would be 90s-mid 2000s, as I read it      around 2008-2011.         The book cover was in first person view, with the main character   holding a weapon, and was a 3D rendered image inside a dungeon or   cave (that was very reminiscent of early 90s CGI animation), though   there may be variants.    It would have been set around when computer   games were mainstream (maybe around the same time as when it might've   been published) It's about a teenage boy who is bored at home, so he   looks through an old family computer to play a game and finds one   that he has never seen before. When he plays it he gets transported   into the game which is a fantasy-medieval setting.    All that I can   remember about the game and story is that the main character is on a   quest, he rides horses a lot, (to the point where he can ride horses   perfectly in real life), at one point he eats food at an ingame   tavern/restaurant? and he tosses the ingame coin currency at the   server because he's in a hurry, (don't know why I remember that bit   specifically haha), and he meets a gnome/dwarf character who is   helping him on his journey, who has a home in an ingame mountain, and   that time goes faster in the game. (For example, if he spent 5   minutes in the real world, 3 in-game days would have passed)   Sometimes as he's playing something stops him from being able the   play the game for a few weeks, maybe a concerned parent because he's   playing too much or there's a family vacation. I remember during the   time when he's not playing the game he's on a farm and knew how to   ride the horses there.    There was a character in the real world who   was the main character's sibling, relative or friend. They were like   a rival and was also very whiny (to some extent). They were on the   farm as well with him and questioned how the main character knew how   to ride a horse when he had never done it in real life.   This is all I can remember but if you need more specific information I can try and answer. It is definitely not Space Demons or Demons Don't Dream. If anyone knows anything remotely similar please mention it, I would really appreciate it. Thank you!!            Additional info:    It is not an online game or virtual reality game. It was a normal PC game, and all other characters are (I think), just NPCs.  He woke up for the first time in a field/farm and was super confused about what happened and where he was, as he had never played the game before. He is asked a super serious question before starting the game up for the first time, like "are you sure you wish to play ___?", and then he is transported into the game.  He frequently exited the game, to continue his IRL life. And sometimes, he would lose access to the family desktop PC for a couple of days at a time, and when he went back everyone in the game (NPCs), thought he had disappeared because he was gone for ages. I think once, his family when on holidays and he couldn't take the PC, for example.  It was a book series, not a single book. In later books he would play the game again months later in the real world, and years and years would have passed in the game world.  The plot involved 2 main warring sides/factions/countries of humans, but there were fantasy/magical races and characters involved. I vaguely remember something about each side having official colours, one red and one blue. I think the main character was on the blue side.   The original question was never correctly answered, so I'm posting again to see if anyone can remember.   Books that were ruled out in original post: Eden's Gate &amp; Heir Apparent. Justify text inside itemize environment with beamer class Difference between pre-increment and post-increment in a loop? Is there a difference in ++i and i++ in a for loop? Is it simply a syntax thing? Ted Shifrin on this site posed an interesting problem to me: show that   $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta &gt; 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta &gt;0$$  I think I've got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma \leq \delta$. Then switch to polar coordinates and get   $$\frac {r^ {\alpha + \beta} (*) } {r^\gamma (1(*) + r^{\delta - \gamma}(*))},$$  where $(*)$ represents some trig stuff that is bounded near zero.   Edit: I need to make sure that $(1(*) + r^{\delta - \gamma}(*))$ is bounded below here.  The other direction is giving me some trouble. It seems we need a lower bound for the fraction (something to force to zero), and I'm not sure where to find one. In particular, I'm not sure what to do with the denominator. Any ideas? Difference between information and knowledge? As far as I think projection of data unto subject mind is information for the subject whereas indentation or impressions accumulated owing to projection of data unto subject mind is knowledge for the subject. I want to know am I right? Or can someone provide a more rigorous answer! Focus existing terminal with `Ctrl-Alt-T` shortcut I tend to use the terminal a lot,  So I wondering if there is a way I can make Ctrl+Alt+T focus the existing terminal if there is one, otherwise create a new terminal? Do spell effects move if the object they are cast on moves? let's suppose that I cast a Silent image of a wall on an airship. Let's suppose that I don't move it with concentration.  Does the image of the wall move with the airship or it will progressively go far? Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$ Problem:   If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|&gt;1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$.   This approach is definitely wrong:    \begin{align} b&amp;=\lim_{x\to 0} \frac{f(ax)-f(x)}{x}\\ &amp;=\lim_{x\to 0} \frac{f(ax)-f(0)-(f(x)-f(0))}{x}\\ &amp;=af'(0)-f'(0)\\ &amp;=(a-1)f'(0) \end{align}  I will show you a case why this approach is wrong:     \[f(x)= \begin{cases} 1,&amp;x\neq0\\ 0,&amp;x=0 \end{cases}\]   $\lim_{x\to0}\dfrac{f(3x)-f(x)}{x}=\lim_{x\to0} \dfrac{1-1}{x}=0$   but $\lim_{x\to0}\dfrac{f(3x)}{x}=\infty$,$\lim_{x\to0}\dfrac{f(x)}{x}=\infty$   Does anyone know how to prove it? Thanks in advance! hot network questions have inconsistent CSS for visited/not visited depending on the site you're on This is a HNQ excerpt from Stack Overflow. Visited questions are objectively darker.    This is a HNQ excerpt from Super User. Visited questions are objectively lighter.    This is a HNQ excerpt from Seasoned Advice. Visited questions are objectively darker.    This is a HNQ excerpt from Puzzling, a beta site. Visited questions are objectively darker, although the difference is hard to spot.    This is a HNQ excerpt from Travel. Visited questions actually have a completely different color.    These are 5 different sites, with 5 different formats for how the HNQ sidebar displays links. Some make visited links darker. Some make them lighter. Some make them a little big darker. some give visited links a completely different color. This is very confusing: If someone visits one of these HNQ questions and wants to continue browsing the HNQ from there, they likely expect that the links in that HNQ segment have consistent layout compared to the HNQ segment from the website they came from.  Is this something worth looking into? I realize that each site has their own look and feel, but an element that appears on every site should be consistent IMHO. People have expectations about how links work, and one of those is that if you click on a link within the same domain (which the HNQ segment arguably is), they behave similary on that page as they do on the page you come from. I am often interested in the installation triggers (postinst, postrm) or certain parts of packages (like /usr/share and /etc). Currently, I am running the next command to retrieve the source code:  apt-get source [package-name]   The downside is, this file is often much bigger than the binary package and does not reflect the installation tree.  Right now, I am downloading the packages through :      Download the actual file   This takes too long for me and as someone who really likes the shell, I would like to do something like the next (imaginary) command:  apt-get get-deb-file [package-name]   I could not find something like this in the . The most close I found was the --download-only switch, but this puts the package in /var/cache/apt/archives (which requires root permissions) and not in the current directory. Why can't we see images reflected on a piece of paper? Assuming i have a line that i want to add to a file without opening an editor.  How could i append this line  alias list='ls -cl --group-directories-first'   to this file  config.fish What does `invalid initialization of non-const reference` mean? Why can "find" not be used in PowerShell? What does find.exe find objectionable about the parameters when it is used in a PowerShell console shell?  These commands work as expected in a cmd.exe shell:  PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; find /i "System.Diagnostics.Process" *.ps1xml FIND: Parameter format not correct PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; find /i "System.Diagnostics.Process" *.ps1xml FIND: Parameter format not correct PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; C:\Windows\System32\find.exe /i "System.Diagnostics.Process" *.ps1xml FIND: Parameter format not correct PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; C:\Windows\System32\find.exe /i "System.Diagnostics.Process" .\DotNetTypes.format.ps1xml FIND: Parameter format not correct How to solve the following congruence equation? $$3n^3+12n^2+13n+2\equiv0,\pmod{2\times3\times5}$$ If $t_n$ be the $n$th triangular number, then  $$t_1^2+t_2^2+...+t_n^2=\frac{t_n(3n^3+12n^2+13n+2)}{30}$$ so if we solve this congruence equation we find the values of $n$ that $t_n$ divides $t_1^2+t_2^2+...+t_n^2$. I am getting this message every time I do something like starting or stopping a service.  perl: warning: Setting locale failed.    perl: warning: Please check that your locale settings:            LANGUAGE = "en_US:en",            LC_ALL = (unset),            LC_MESSAGES = "en_US.UTF-8",            LANG = "en_US.UTF-8"        are supported and installed on your system.    perl: warning: Falling back to the standard locale ("C").    locale: Cannot set LC_CTYPE to default locale: No such file or directory    locale: Cannot set LC_MESSAGES to default locale: No such file or directory    locale: Cannot set LC_ALL to default locale: No such file or directory    (Reading database ... 21173 files and directories currently installed.)   Removing bind9 ...    * Stopping domain name service... bind9                                        [ OK ] Processing triggers for man-db ...    locale: Cannot set LC_CTYPE to default locale: No such file or directory    locale: Cannot set LC_MESSAGES to default locale: No such file or directory    locale: Cannot set LC_ALL to default locale: No such file or directory      How do I fix this error ? The axiom of choice asserts that for any set $X$ there exists a function $f : (2^X − \{\emptyset\}) \rightarrow X$ such that for any nonempty $A \subseteq X$, $f(A) \in A$. Show that this is equivalent to the assertion that every relation includes a function with the same domain. I am given this problem in my assignment. My approach is following. I have a set $X$. Then it has a choice function $g : \mathcal{P}(X) \setminus \left\{\emptyset\right\} \rightarrow X$. Suppose $X$ has a relation $R \subseteq X \times X$. $dom(R) \subseteq X$. Then the required function $f : dom(R) \rightarrow X$ is $$f(x) = g(\{y\ |\ (x, y) \in R\})$$ Conversely, suppose $X \supseteq Y$ is a non-empty subset. Let $$R_Y = \{(x, y)\ |\ y \in Y, x \in Y, (p, q) \in R_Y \Rightarrow p = x\}$$ Then there is a function $f_Y : dom(R_Y) \rightarrow X$. But $dom(R_Y)$ is singleton from the definition of $R_Y$. Then $im(f_Y)$ is also singleton. We define our choice function as $f = \{(Y, y)\ |\ y \in im(f_Y)\}$ Are my arguments correct?  Edit.1: what I tried to do for converse is, for any subset $Y$, I construct a relation $Y_1 \times Y$ where $Y \supseteq Y_1 = \{y\}$. Then there is a function $f_Y : Y_1 \rightarrow Y$. Then if $f$ is my choice function I define $f(Y) = f_Y(y)$. May be I should define $R_Y$ as $$R_Y = \{(x, y)\ |\ y \in Y; x \in Y; (p, q), (p', q) \in R_Y \Rightarrow p = p'\}$$ I understand there may be multiple $R_y$'s. But can't I define a singleton set as $\{s\in S\ |\ a, b \in S \Rightarrow a = b\}$? This can be $\{s\}$ $\forall s \in S$. But isn't it a valid specification?  Edit.2: This is the final solution with the help from @Asaf.  If we assume axiom of choice then for any $X$ there is a choice function $f : \mathcal{P}(X) \setminus \left\{\emptyset\right\} \rightarrow X$ such that $f(A) \in A$. Suppose $R$ is a relation between $P$ and $Q$. Then we have a choice function $f_Q$ for $Q$. Let $f_R(p) = f_Q(\{q\ |\ pRq\})$. The function $f_R : P \rightarrow Q$ and $dom(f_R) = dom(R)$. So this is our required function.  Conversely, if every relation includes a function with same domain. Let $R$ be a relation between $\mathcal{P}(X) \setminus \left\{\emptyset\right\}$ and $X$ such that $aRb \Rightarrow b \in a$. Now $R$ includes a function $f : \mathcal{P}(X) \setminus \left\{\emptyset\right\} \rightarrow X$. Then $xRf(x) \Rightarrow f(x) \in x$. So $f$ is the choice function for $X$. If my question gets migrated to another site, is it possible somehow to reverse it back to the original site? Continuity of absolute value of a function Let $f(x)$ be a continuous function. Prove that $\left|f(x)\right|$ is also continuous.  Is it correct to say that, by the reverse triangle inequality, $\left|f(x)-f(c)\right| \geq \left|f(x)\right|-\left|f(c)\right|$ in all cases, so we will always have that for all $\left|x-c\right|&lt;\delta$ implies $\left|f(x)-f(c)\right|\leq \epsilon$?  I am not sure if my solution adequately uses the functional form, since I just used the equation and not my actual function. Please help with the proper solution! In the LoTR movies (not sure if the books are the same) but when Gollum lost the ring to Bilbo decades ago in The Hobbit, he appeared to look the same and even move the same in The Fellowship of the Ring. Bilbo on the other hand, seemed to have aged very rapidly when he passed the ring to Frodo. When Frodo met up with Bilbo in Rivendell shortly after his quest began, Bilbo appeared much older and even more much slower.  I realize that Gollum had the ring for much longer than Bilbo, but wouldn't him losing the ring have the same effect as it had on Bilbo?