Here goes:    imagine a spacecraft with engines shut off in a free fall towards some celestial body x. let's say the distance to surface is $100 000$ $\text{meters}$.  initial vertical velocity is $500$ $\text{m/s}$.  gravitational pull experienced by the body is obviously given by $F=G \frac {m_1  m_2}{r^2}$. acceleration of said craft is calculated as $a = \frac{F}{m}$.   Given constant $F$ and therefore $a$, impact velocity and time are easy to calculate, but how exactly do I solve this, considering  that gravitational pull and  acceleration are constantly changing as we approach the surface?  Key question in  is "To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0?"  The answer by David H involves the following formula $v_f = \sqrt{\frac{2k}{x_0} + v_0^2}$  In my case x goes from 100000 to 0 m (surface), in other words, there is a clearly defined endpoint which is not equal to infinity. Therefore I do not see how David's formula is applicable in my situation.
A particle experiences an acceleration described by  $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant.  To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0?  If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$  = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$  = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$  How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration)  Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles  What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0?
If $f \colon [a,b] \rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$ If $f \colon [a,b]\rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$.  I think that I have to proceed with the Mean Value Theorem of Integrals... How do I handle the window close event in Tkinter? There is a button in the "basic math assistant" which wraps highlighted text in chosen brackets. Is there no keyboard command for this? Lately I see a blue screen of death a lot and want to know why it appears. Where should I gather the information so the problem can be diagnosed best? Couldn't find anything in C:\Windows\Logs. Using biblatex and Biber, I want my bibliography to say "Ph.D. thesis" (with periods) rather than "PhD thesis" (without). MWE:  \documentclass{article} \usepackage{biblatex} \addbibresource{refs.bib} \begin{document} Hello \cite{testphd} \printbibliography \end{document}   and refs.bib:  @thesis{testphd,   title       = {Thesis Title},   author      = {Surname, Firstname},   date        = {2010},   institution = {Faculty of X, University of Y},   type        = {phdthesis},   location    = {Somewhere}, }   Output:    What's the easiest way to do this? I don't want to change the .bib file ideally. I recently saw a question about a problem (or possible bug) with some product on Stack Overflow. At first, I wanted to suggest that the asker post his question to that product's support forum.  When searching for the URL of that forum, I noticed that the product developer closed its support forums and now advises users to post all their questions on Stack Overflow instead.  Is this within the intended use cases of Stack Overflow, or is this abuse? Delete with Join in MySQL Here is the script to create my tables:   CREATE TABLE clients (    client_i INT(11),    PRIMARY KEY (client_id) ); CREATE TABLE projects (    project_id INT(11) UNSIGNED,    client_id INT(11) UNSIGNED,    PRIMARY KEY (project_id) ); CREATE TABLE posts (    post_id INT(11) UNSIGNED,    project_id INT(11) UNSIGNED,    PRIMARY KEY (post_id) );   In my PHP code, when deleting a client, I want to delete all projects posts:  DELETE  FROM posts INNER JOIN projects ON projects.project_id = posts.project_id WHERE projects.client_id = :client_id;   The posts table does not have a foreign key client_id, only project_id. I want to delete the posts in projects that have the passed client_id.  This is not working right now because no posts are deleted. Flirtatious Primes Here's a possibly interesting prime puzzle. Call a prime $p$ flirtatious if the sum of its digits is also prime. Are there finitely many flirtatious primes, or infinitely many? Our DM keeps interrupting our long rests In the latest campaign, we're still around the third level, but every single night we've spend in the wild so far, we were interrupted by hard combat encounters (as in DMG-worded "hard"), forcing us to use our resources (like HP, spell slots, Ki, once-per-day stuff) and not restoring any of those lost the previous days. I feel like the , as it lasts four to six rounds, so we usually say that we'd like to resume our rest but the answer is always "you've been interrupted, your rest is over, you can't restart any until dusk". So the only real long rests we could have always were in town.  We've always used watch-turns, and being interrupted in the first night of the campaign, we now always spend one hour to find a safe place to rest and we make that intent very clear to the DM. But yet, kobolds, goblins, ghouls and gnolls always find us in the middle of the night, but never when we travel for ten hours during the day (which is comparable to "fast travel" in his own word). Once on the second day in the wild, we were faced to a deadly encounter (as in DMG-worded "deadly") which we survived only by fleeing. And we're kind of forced to get back to the deadly encounter, which is more than one day away from the nearest town.  We're five players and I'm the only one with PnP RPG experience... from ten years ago. I don't remember being this resourceless. We asked if it was a survival campaign and the answer was that it was only a normal introduction campaign found on the Internet that he spiced up a bit. We've also asked if we did anything wrong and the answer was that we followed the normal story.  The DM is mastering his second campaign (the first one is still ongoing, but with veteran players). The DM also plays a lot of survival campaigns as a player.  Is this situation normal? If yes, why? If not, what should I say to the DM?  Note that I'm fully aware of the "long rest" rule as written on page 186 of the PHB. This is why I encouraged my team to say that we resume our rest. (thanks Bloodcinder) When I parent mesh to an armature (automatic weights), the mesh "jumps" forward a bit. If I select armature and switch to edit mode it "jumps" backward a bit.   Why is it happening and how can I fix it? I already applied location, rotation, and scale to the mesh and armature, but it didn't help.  Thank you in advance! I have a machine running Ubuntu which I SSH to from my Fedora 14 machine. I want to forward X from the Ubuntu machine back to Fedora so I can run graphical programs remotely. Both machines are on a LAN.  I know that the -X option enables X11 forwarding in SSH, but I feel like I am missing some of the steps.  What are the required steps to forward X from a Ubuntu machine to Fedora over SSH? If $z_n \to z$ then $(1+z_n/n)^n \to e^z$ We are dealing with $z \in \mathbb{C}$. I know that $$ \left(1+ \frac{z}{n} \right)^n \to e^{z} $$ as $n \to \infty$. So intuitively if $z_n \to z$ then we should have $$ \left(1+ \frac{z_n}{n} \right)^n \to e^{z}. $$ If $z_n \in \mathbb{R}$ I would be happy writing $$ \exp \left(n \log\left(1+ \frac{z_n}{n} \right) \right) = \exp \left(z_n \frac{\log\left(1+ \frac{z_n}{n} \right)-\log(1+0)}{\frac{z_n}{n}-0} \right) \to \exp(z \cdot 1) $$ where here we are using the definition of derivative. But if $z \in \mathbb{C}$ the logarithm is multivalued and I'm not sure that the same calculation is allowed. Is there a different way to show this for complex $z$? I need to install a new TeX/LaTeX/ConTeXt package on my computer.  I'm running Ubuntu/Fedora/Slackware/Some-Version-Of Linux.  What are the possible ways to do this?    Note:&nbsp;&nbsp; We get a number of questions that essentially reduce to this one; it was felt a good idea to have one question-and-answer that can cover all the standard methods.  See  for the discussion.  The question-and-answer are CW so that more information can be easily added.  If you have information to add, please do so.  If you are unsure, leave a comment here or (preferably) in .  Comments on the question and answer will be periodically removed to ensure that the information is clear. How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated. Finding the variable name passed to a function std::auto_ptr to std::unique_ptr Examples of provably${}^n$ unprovable statements Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement is provable, then it is provably provable). Let $T_1=T$ and $T_{n+1}=T_n+{\sf Con}(T_n)$. (We need stronger theories if we want to prove unprovability.) Given this, simply by considering the truth values of the relevant statements, we get the following classification of statements by unprovability:   $T\vdash A$ ($A$ is provable) $T_2\vdash T\not\vdash A$ ($A$ is provably unprovable) $T_3\vdash T_2\not\vdash T\not\vdash A$ ($A$ is provably unprovably unprovable)   &nbsp;&nbsp;&nbsp;&nbsp;...  &nbsp;&nbsp;&nbsp;&nbsp;$\omega$. For all $n$, $T_n\not\vdash\dots \not\vdash T_2\not\vdash T\not\vdash A$ (we can't say anything about $A$'s provability)  We can refine this hierarchy by also considering the status of $\lnot A$. Let's call a class $m,n$ statement one such that $A$ is at level $m$ above and $\lnot A$ is level $n$ in the list. Since $T\vdash\lnot A$ implies $T_2\vdash T\not\vdash A$, the only nonempty classes at level $1$ are the class $1,2$ statements (provable statements) and the class $2,1$ statements (refutable statements). Independent statements are in class $2,2$. The class $1,1$ is nonempty iff the theory is inconsistent.  My question is: Are there any (preferably "natural") statements that fall further into this hierarchy? I know many examples of independent statements, like the continuum hypothesis or the axiom of choice relative to $\sf ZF$, or the Paris–Harrington theorem relative to $\sf PA$. But I know no examples of even any class $2,3$ statements. I expect there may also be natural examples of $\omega,\omega$ statements. My guess is that all the classes $m,n$ with $m,n\ne1$ are nonempty, but examples elude me. How can I convert from a SID to an account name in C# I have a C# application that scans a directory and gathers some information.  I would like to display the account name for each file.  I can do this on the local system by getting the SID for the FileInfo object, and then doing:  string GetNameFromSID( SecurityIdentifier sid ) {     NTAccount ntAccount = (NTAccount)sid.Translate( typeof( NTAccount ) );     return ntAccount.ToString(); }   However, this does not work for files on a network, presumably because the Translate() function only works with local user accounts.  I thought maybe I could do an LDAP lookup on the SID, so I tried the following:  string GetNameFromSID( SecurityIdentifier sid ) {     string str = "LDAP://&lt;SID=" + sid.Value + "&gt;";     DirectoryEntry dirEntry = new DirectoryEntry( str );     return dirEntry.Name; }   This seems like it will work, in that the access to "dirEntry.Name" hangs for a few seconds, as if it is going off and querying the network, but then it throws a System.Runtime.InteropServices.COMException  Does anyone know how I can get the account name of an arbitrary file or SID?  I don't know much about networking or LDAP or anything.  There's a class called DirectorySearcher that maybe I'm supposed to use, but it wants a domain name, and I don't know how to get that either - all I have is the path to the directory I'm scanning.  Thanks in advance. Windows 10 update broke my MBR / GRUB Direction of $d\mathbf{l}$ A solid sphere has charge $q$ and radius $R$. Find the potential at a point a distance $r$ from the center of the sphere where $r&gt;R$, using infinity as the reference point.  My attempt:  From Gauss' theorem we may deduce that $\displaystyle\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r'^2}\hat{\mathbf{r}}$ where $r'$ is the distance of an arbitrary point from the center of the sphere provided $r'&gt;R$.  $V=-\int_\infty^\mathbf{r}\mathbf{E}\cdot d\mathbf{l}$  My question: What is $d\mathbf{l}$? Since we are traversing in the direction opposite to $\hat{\mathbf{r}}$, I think it should be $-dr'\hat{\mathbf{r}}$. But when I use it to find $V$, I get a sign error. Please help!