Consider a particle of mass $\text{m}$ fired from below into a bob of mass $\text{M}$ of a long, simple pendulum. The particle remains inside the bob and the bob rises through a height of $1.8\text{ m}$. What was the speed of the particle just before striking the bob?  I took the bob, the particle and the earth to be in a system and reasoned that their total mechanical energy must be conserved, since no external forces act on the system and all internal forces are (I think) conservative. Then, I set my reference line for measuring potential energy to be at the level of the bob before the collision. The kinetic energy of the system just before the collision must be equal to the gravitational potential energy at the highest point of the bob's motion. $$\begin{align}\frac{1}{2}mv^2&amp;=(M+m)gh \\ v &amp;= \sqrt{\frac{2(M+m)gh}{m}} \end{align}$$ However, my textbook uses the principle of conservation of linear momentum and the laws of kinematics:  If $\text{V}$ is the velocity of the combined mass after the collision, $(M+m)V=mv$. This gives: $$V=\frac{mv} {M+m}$$ As the string becomes loose and the bob goes up with a deceleration of $\text{g}$, coming to rest at a height $\text{h}$, we can use the equation $v^2=u^2+2ax$. $$h=\frac{V^2}{2g}=\frac{m^2v^2}{2g(M+m)^2}$$ $$v=\frac{M+m}{m}\sqrt{2gh}$$  Why did we get a larger answer this time? My intuition is that be conserving the mechanical energy of the system in the first method, I ignored possible energy losses in the deformation of the bob. Is my intuition correct?
I am stuck in a question where energy conservation is failing and momentum conservation is correct. I think I might be doing something wrong, that's why I'm asking this question. The problem is as follows:     A bullet of mass $25\,g$ is fired horizontally into a ballistic pendulum of mass $5.0\,kg$ and gets embedded in it. If the centre of pendulum rises by a distance of $10\,cm$, find the speed of the bullet.   (H.C. Verma, Centre of Mass, Q47)     Let the mass of the bullet be $m=25\,g$. Let the mass of the pendulum be $M=5\,kg$. Let the peak height be $h=0.1\,m$. Let the initial and final velocities be $u$ and $v$ respectively.  Method 1: (Momentum Conservation)  $$mu=(M+m)v \implies v=\frac{mu}{M+m}\\ \text{Also, }\frac{1}{2}(M+m)v^2=(M+m)gh\\ \implies u^2=2\biggl(\frac{M+m}{m}\biggr)^2gh\\ \implies u=201\sqrt2 \text{ m/s}$$  Method 2: (Energy Conservation)  Since both the masses move together after the collision, and because the velocity at highest point is null, therefore: $$\frac{1}{2}mu^2=\frac{1}{2}(M+m)v^2=(M+m)gh\\ \implies u^2=2gh\biggl(\frac{M+m}{m}\biggr) \implies u=\sqrt{402}$$    To summarize myself, I am curious about the following: Why do the results differ? Shouldn't the energy be conserved as well as the momentum? Since it should be, then what is the flaw in my calculations?
Why do I have different IP addresses on different websites? My wireless router shows the IP address is 113.46.75.*, which some websites like ip138.com, www.ip.cn show as my IP.   But other websites, like googling for "what is my ip", www.whatismyip.com, www.whatsmyip.org, show that my IP is 121.18.126.*.   I am very curious why this happens and how is it possible.   So, why do different websites show that I have different IPs?  Thanks all!  Plus: I changed my IP address several times (restarting the modem) and all addresses is like 113.46.*.*, but my IP address does not change when googling for "what is my ip".  My ISP is a small company, so I think it is true that when visiting some websites, I am NATted or proxied to some IP addresses which have access to those websites -- not to avoid firewall though, since I am still blocked from some websites.  But I still want some more information or resources to support our guess. Thank you! How can I prove the following?  $\gamma (t)$ is unit speed, $\dot \gamma(t) \not= 0 \Rightarrow \ddot \gamma(t)$ is perpendicular to $\gamma(t)$   I don't really see where a problem would arise when $\dot \gamma(t)=0$ would cause such a problem Children's fantasy series with a symbol consisting of a circle divided into quarters I read this book series as a kid back in the late 1970s, so it's an old series - and my memories are similarly old, and vague.   I know it's a fantasy series. I think there are 4 books in the series. Or perhaps 7.  There's a central boy character who has to get a collection of magical objects (every fantasy needs a quest!), each of which represents a different magical 'element': wood, stone, fire, etc. I definitely remember there was a wooden object. He lives in our normal world. The whole series takes place "here &amp; now". I have a feeling he's a "chosen" person of some kind... But, the strongest memory I have is of a symbol: a circle which is divided into quarters (a vertical bar and a horizontal bar).  I know it's vague, but does anyone recognise this? What is $n!$ when $n=0$? Basically on my site I have a bunch of song lyrics and in a static page I want to show them listed by album  and album position of the song and I have two custom fields with that data, sd_album_title and `sd_album_track_n. Plus some of the songs don't have an album yet, so both values are void, and I wish to list those ones at the foot of the others, sorted by alphabetical order.  The latest thing I tried at least to sort everything by the two field is this one:  $key = 'sd_album_title'; $themeta = get_post_meta($post-&gt;ID, $key, TRUE);  add_filter( 'pre_get_posts', 'my_get_posts' ); function my_get_posts( $query ) {     if ($themeta != '') {         $query-&gt;set( 'post_type', 'lyric' );         $query-&gt;set( 'meta_key', 'sd_album_title' );         $query-&gt;set( 'orderby', 'meta_value' );         $query-&gt;set( 'order', 'ASC' );         $query-&gt;set( 'meta_query', array(               array(                     'key' =&gt; 'sd_album_title'               ),               array(                     'key' =&gt; 'sd_album_track_n',                     'value' =&gt; 'target_value',                     'compare' =&gt; '&gt;=',                     'type' =&gt; 'numeric'               )         ));     }     return $query; }  $getlist= 'my_get_posts'; $posts = get_posts( $getlist );   And my intention was to put an else at the end of it with an array ordered by title for the other songs, but it returns nothing as it is. Could something like this work with some corrections or am I on a totally wrong path? I found this math "problem" on the internet, and I'm wondering if it has an answer:     Question: If you choose an answer to this question at random, what is the probability that you will be correct?      a. $25\%$      b. $50\%$      c. $0\%$      d. $25\%$   Does this question have a correct answer? How did Harry's blood being inside Voldemort save him? There are, of course, an incredible amount of theories on how Harry Potter survived. Problem is that the most important one, the one that is canon, is the one I don't get: how Harry's blood inside Voldemort saves him?  Because all that I have read so far, leaves me with questions. I understand why Voldemort wanted Harry's blood, but why does it save Harry? Why is that necessary to keep the sacrifice alive, while it's still inside Harry? And if Harry lost it at 17 (I have seen some different ideas on this as well), why does Voldemort still have it at 69 (give or take a few)?  To clafiry, the parts I'm talking about:     "For a fleeting instant, Harry thought he saw a gleam of something like triumph in Dumbledore's eyes"    (GoF, right after Harry tells Dumbledore that his blood now goes through Voldemort.)  And:     "He took your blood believing it would strengthen him. He took into his body a tiny part of the enchantment your mother laid upon you when she died for you. His body keeps her sacrifice alive, and while that enchantment survives, so do you"   (Albus Dumbledore in DH)  I just don't get why the blood transfusion thing suddenly gave Harry the chance to survive the whole thing, and it's never really been satisfyingly answered, not to my knowledge at least, as every answer I've read raises more questions. String representation of an Enum Recently, I accidentally deleted a partition off of my external hard disk. I had a copy of Minitool Power Data Recovery left over when the same drive had become RAW about a year ago. When I selected lost partition recovery, it immediately identified my deleted partition and allowed me to access and recover the files that were there.  My question is this. Is there any way to change the deleted partition back to NTFS without having to have a drive that you need to recover the files to and then move them back after formatting, as I don’t have any space left on either the computer or on my other external hard drive, so that I don’t need to buy a new one. The files on the disk were really important.  I am using Windows 7 Home Premium.  I appreciate any help in solving this crisis. I want to create a simple figure where images are connected with arrows. Here is what I did:  \begin{tikzpicture} \node[](A) {\includegraphics[width=0.2\textwidth]{a}}; \node[right=1mm of A](B) {\includegraphics[width=0.2\textwidth]{b}}; \draw[-&gt;] (A) to (B); \end{tikzpicture}     But I want the arrow to start and end right at the images with no space between. How can I do that? If $F:M\to N$ is a smooth embedding, then so is $dF:TM\to TN$. Do I have 32-bit or 64-bit? Vote for Tag Synonyms Directly on the Page Before:  After:   The pros:  Speeds up the tag synonym voting process. Easier to locate (as opposed to navigating to each individual tag wiki page). Encourages users to vote on more than one at a time.  The cons:  Serial up/downvoting. But this really isn't even an issue because, (1) only experienced users can vote on tag synonyms, and (2) more than one vote is required to approve a tag synonym, so the occasional serial up/downvoter won't significantly impact the overall process. Troubleshooting "program does not contain a static 'Main' method" when it clearly does...? Tips to extend battery life for laptops and notebooks This question is present as a matter of historical interest.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will be closed per the .   It is possible to greatly extend the usage time of a laptop or netbook running on battery by disabling various services and installing various packages.   What tricks or tips do people have for getting an extra hour or two out of their batteries? Use Outgoing SSH to receive incoming SSH My ISP does not allow me to have a static IP address. As such I am unable to remotely SSH in my home machine. Is there a way I could SSH from my home machine to a server and SSH into the server from my remote laptop and bridge the two connections?  Cheers! Shortcut in Windows 7 to switch between same application's windows, like Cmd + ` in OS X Is there a shortcut in Windows 7 to switch between windows in a given application only? I very much like this feature in OS X; what is its equivalent in Windows? How do I use an NBT editor to modify item properties? What forces affect the value of my ETF? Where to insert the exclamation mark Where would I insert an exclamation mark and a comma in the below sentence?     "Look at that boy" he exclaimed.