Displacement and Pressure Wave
Pressure variation in a sound wave
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. How to Calculate the time of Charging and Discharging of battery? How to disable notifications for a single mailbox in Mail? When managing multiple mailboxes, is it possible to disable the notifications for certain mailboxes? Extended support for Ubuntu 12 Not sure if there is a better communication mechanism for asking this question. Ubuntu 12 is EOL April 2017. Do corporations have any option to extend support on 12?  Thanks Is there a difference between DataTable.Clear and DataTable.Rows.Clear? I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. Distorted Cabin & Oily Sheen: What is going on in this paragraph? Nautilus 3.6 doesn't have a status bar? In 12.04, Nautilus showed the amount of remaining space in the status bar , but as of 13.04 the status bar is gone.  gsettings set org.gnome.nautilus.window-state start-with-status-bar true   turned out "no such key".  Is there any way to bring back the status bar? Why couldn't River Song's memories bring back The Doctor in the rebooted universe? In the season 5 finale of Doctor Who, in the wedding ceremony of Amy, River Song showed up outside the door. Moments later, we discovered that she gifted Amy her blank diary which was supposed to contain information about The Doctor. It means River Song remembered The Doctor.  Memories of The Doctor was able to bring The Doctor back to existence in the rebooted Spacetime. Why couldn't River Song's memories work? How do the comment replies (aka &quot;pinging users&quot;) work?  Who can I send an @reply to? How do I respond to a specific user when entering my comment? Will they be notified? What do people mean when they talk about &quot;pinging&quot; another user?  Related: How to create a virtual audio device and stream audio input with it Here's my concrete problem:  I am broadcasting a game via XFire and it uses the Windows audio device to capture any audio I receive. As I am broadcasting, other users who watch the video stream are communicating with me over Skype, and they hear themselves back within the video stream and it is entirely logical since I am broadcasting the audio I hear.  What I want to do is create another audio device within Windows and redirect (pipe) ONLY the audio input from that game and not the input reveived from Skype. I would then tell XFire to use that newly created "virtual" audio device to broadcast and therefore my partners won't hear themselves back.  Is it possible to create another audio device and redirect only wanted input streams to it? PS4 Controller not connecting to PS4 and won't reset? Why Was It 'Essential' That Voldemort Kill Harry Potter? ‘So the boy ... the boy must die?’ asked Snape, quite calmly.   ‘And Voldemort himself must do it, Severus. That is essential.’   Deathly Hallows - page 551 - UK Hardcover - chapter 33, The Prince's Tale   Dumbledore tells Snape it is 'essential' that Voldemort be the one to kill Harry, I'm assuming in order for the piece of Voldemort's soul in Harry to be properly killed as well.  But why?   Hermione destroyed the Hufflepuff cup; Ron destroyed the Slytherin locket; Neville killed Nagini; Harry destroyed the diadem and Tom Riddle's diary. So the Horcruxes were not immune to destruction at the hand of someone other than Voldemort. And as it ended up, Harry himself wasn't even a Horcrux, but rather just an unknowing host to a parasitic bit of Voldemort's soul.   Why was it 'essential' that Voldemort be the one to kill Harry in order for the piece of Voldemort's soul to die? Unfortunately MyApp has stopped. How can I solve this? Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? In PHP, how do you change the key of an array element? I have an associative array in the form key =&gt; value where key is a numerical value, however it is not a sequential numerical value. The key is actually an ID number and the value is a count. This is fine for most instances, however I want a function that gets the human-readable name of the array and uses that for the key, without changing the value.  I didn't see a function that does this, but I'm assuming I need to provide the old key and new key (both of which I have) and transform the array. Is there an efficient way of doing this? How can I restore configuration files? I created some linestring features in EPSG:32632 (WGS 84 / UTM zone 32N) exactly 10 000 m long.  QGIS calculates all line lengths (horizontal, vertical, diagonal) as 10 001.9 m. The measure tool explains "Project CRS transformation is turned on and ellipsoidal calculation is selected. The coordinates are transformed to the chosen ellipsoid (WGS84), and the distance is calculated in meters."  When I set the ellipsoid to "None / Planimetric" in project properties, I get the expected results (10 000 m).  I don't understand the phrase "coordinates are transformed to the chosen ellipsoid (WGS84)", is it a transformation to unprojected degrees (4326?)? Since 32632 already is based on WGS84, is there anything to transform? If there is a kind of "great circle" calculation involved, I would have expected lengths can only get smaller.  Is QGIS's calculation correct/meaningful, is it plain wrong or do I see rounding errors?  This are the geometries I tested:  LineString (370000 5615000, 370000 5625000) LineString (366464.46609406732022762 5616464.46609406732022762, 373535.53390593267977238 5623535.53390593267977238) LineString (365000 5620000, 375000 5620000) LineString (373535.53390593267977238 5616464.46609406638890505, 366464.46609406638890505 5623535.5339059317484498)   Edit as addition to MappaGnosis's answer: I forgot that in UTM , this seems to be the simple answer for the title question. I have been running  over some C# code, and it keeps reporting that my using directives should be inside the namespace.  Is there a technical reason for putting the using directives inside instead of outside the namespace? Can we theoretically balance a perfectly symmetrical pencil on its one-atom tip? I was asked by an undergrad student about this question. I think if we were to take away air molecules around the pencil and cool it to absolute zero, that pencil would theoretically balance.  Am I correct?  .