When we project an object vertically upwards its velocity at maximum height consider to be zero but still, it has acceleration due to gravity i.e $9.8\mathrm{m/s^2}$. But mathematically we know that acceleration is the first derivative of velocity with respect to time $(a=dv/dt)$. So mathematically if velocity is zero there can not be any acceleration but this case contradict this equation. We can not even say that its direction is changing as the direction has no meaning without magnitude. So how this situation can be explained mathematically?
When I shoot a ball vertically upward, its velocity is decreasing since  there is a downward acceleration of about $9.8\,\mathrm{ms}^{-2}$.   I have read that at the top most point, when $v = 0$, the acceleration is still $9.8\,\mathrm{ms}^{-2}$ in the downward direction where $v=0$. That is, the acceleration is still the same.  But at the highest point, the ball is stationary, so it is not even moving. How can it accelerate?
Deduction of $\forall y \exists x (x=y)$ Give a deduction of $\forall y \exists x (x=y)$  My thinking is I can prove $\exists x(x=y)$ then use generalization theorem， which is equivalent to prove $\neg \forall x \neg (x=y)$ but I got stuck proving this... thanks in advance  Sorry that I did not specify what deduction system that I want to use here, it is the deduction sequence  where ai is either Axiom or original hypothesis or obtained from MP Evaluating string "3*(4+2)" yield int 18 Is there a function the .NET framework that can evaluate a numeric expression contained in a string and return the result? F.e.:  string mystring = "3*(2+4)"; int result = EvaluateExpression(mystring); Console.Writeln(result); // Outputs 18   Is there a standard framework function that you can replace my EvaluateExpression method with? How to reduce the spacing between responses in the alterqcm package? This question has a continuation here   The alterqcm package is no longer maintained by Alain Matthes. Two other developers maintain it. I wrote to them 15 days ago by email to report the error reported here:  I have not received a reply. I am trying to reduce the spacing between the lines of the QCM as indicated in  by @Werner and it does not work. In addition, by using the David Carlisle's code given in , I get an X instead of a square.  Full code of the screenshot: \documentclass[12pt]{article} %\usepackage[utf8]{inputenc} %\usepackage[upright]{fourier} \usepackage{etoolbox}% http://ctan.org/pkg/etoolbox \usepackage[english]{alterqcm} \def\square{X} % &lt;---------- David Carlisle comment \def\nogreekalph{??} % &lt;---- David Carlisle comment \parindent0pt \begin{document} % Original display \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \bigskip  % Smashed display !!!!!! Dont work now  {\begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {\smash{Proposition 1}},  {\smash{Proposition 2}},  {\smash{Proposition 3}}} \end{alterqcm}}  \bigskip  % Global smashed display !!!! Dont work now  \makeatletter \patchcmd{\aq@prop}{\vbox{#1}}{\vbox{\smash{#1}}}{}{}% patch \aq@prop \makeatother \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \end{document}  Is there a solution? Get Response header when 401 occurred How do I use an NBT editor to modify item properties? Why do I have to enter my password to log in? Using the Stack Exchange login, why do I have to enter my password again and again to log in? Wasn’t the whole purpose of OpenID to do away with this?  This doesn’t even only happen when visiting a new Stack Exchange site – it also happens for sites where I already have an account but which I don’t visit regularly. Here’s what I mean:    Of course I don’t remember my password (it’s a long random string). And so I have to go to my password manager and manually copy the password from openid.stackexchange.com’s account into the login form. – And incidentally, this is what my password manager looks like now:    (And, by the way, those are the only entries in my keychain so far, but by far not the only SE sites where I have an account.)  … I feel that this defeats the purpose of site-wide login and OpenID. In the olden times when MyOpenID was still working properly, I’d simply enter my OpenID on the login form, and the software would take care of the rest – namely, redirect to the OpenID service, (automatically) verify that I was authorised to use it, and redirect back to the original website, having logged me in. It was literally one button click for me. It’s substantially more work now, and I’d like to understand why this regression is necessary.  (Note: of course once a site’s password has landed in my password manager I (probably?) won’t have to enter it again. But still: wasn’t site-wide login and OpenID supposed to solve this already?) How do you prove the follwing $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$ how do you prove this inequality? $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$ At first glance I thought the triangle inequality would be useful but it's not form what I see. If I raise everything to the power of n I get the following: Possibility 1 (n uneven) : $|x + ... -y| \le |x-y|$ I guess I would have to prove that ... $\le$ 0 but I don't know how to do that. Possibility 2 (n even) : $|x + ... +y| \le |x-y|$ Same as above but now I have to prove that ... $\le$ -y. Am I even on the right track here? Any help would be appreciated. Thanks in advance. If I want to solve the equation $4x = e^x$, is there a way to solve for $x$ without graphing and looking for intersection? What are Zombies and what causes them? Are there Zombie processes and Zombie objects? Our son is about to turn 3 months but he is yet to sleep on his own (as in we put him to sleep, put him down on the crib then he sleeps for few hours). He only sleeps when he is carried during the day, the moment we put him down he wakes up crying. At night, he sleeps with us on the bed where he is right next to his mommy. We blame ourselves for letting this happen but now my wife can't do anything during the day but carry him.  What can we do to break him out of this habit? Any pointers would be great appreciated. Trailing Sum Query Difference between HTMLCollection, NodeLists, and arrays of objects Are players allowed to pass a penalty kick? Comment character counter does not trim, while the server does If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$. If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.   I want to use induction to prove this:  It is trivial when $G=p(n=1)$, assume $G=p^n$, $N$ is a normal subgroup in $G$ with order $p$. Since $G$ is p-subgroup, $G$ has a normal subgroup $H$ with order $p^{n-1}$,according to assumption, $N \subset C(H)$. But I don't know how to go on, is $C(H)=C(G)$?Can induction work in this case? delay JQuery effects I want to fade out an element and all its child elements after a delay of a few seconds. but I haven't found a way to specify that an effect should start after a specified time delay. I use to work at night and it is eye-hurting to watch at lightbulb-bright screens (I've got two - laptop's built-in panel and an external 18-inch CRT) in a dark environment. So I adjust my screens to be darker.  Can I do this a software way instead of pressing monitors' buttons that many times every time? How would we tell antimatter galaxies apart? Given that  galaxies are theoretically possible, how would they be distinguishable from regular matter galaxies?  That is, antimatter is equal in  and all properties, except for the opposite reverse charge of the particles, identical to regular matter. Hence a star composed of antimatter hydrogen would fuse to anti-helium in an analogous way to our own Sun, and it would emit light and radiation at the same wavelengths as any regular matter star and would cause the same gravitational forces for planetary systems to form as in any other star system.  Hence, what would be a telltale sign if you were observing a galaxy made up entirely of antimatter?  Also, is there any evidence for that half of all galaxies are not made of antimatter -- while general theories currently assume that there is an imbalance of matter over antimatter in the universe, then what is the rationale for not assuming that there is in fact an even balance between the two? 'An avalanche of mail'? - it should be 'mails', shouldn't it? I came across this example on WordWeb while going through the meaning of the word ''.     the program brought an avalanche of mail (Just like "a group of man men").   I'm quite sure it should be mails. Kindly confirm. I'm fairly certain this has happened to all of us at some point: You go into a room to get something, but once you're there, you can't remember what you intended to get. It seems like a specific, common phenomenon like this must have a name, but I can't think of it. Does anyone know a name (either formal or slang) for this?  I don't want a general term like forgetfulness or absent-mindedness, which could apply to all sorts of forgetting, or a term for not being able to remember a certain word. I'm looking for a word or phrase that describes this one specific situation of forgetting what you need to get as soon as you go to get it.