Wikipedia:Reference desk/Archives/Science/2024 February 27

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February 27

Bulletin of the Seismological Society of America in online libraries?

I'm trying to make the cited sources in White Wolf Fault as accessible to readers as possible but I haven't been able to find this journal in the Internet Archive, JSTOR or Wiley. HathiTrust only has the issues from 1911 to 1926. The Seismological Society of America's website has has the issues cited, but it requires library or institutional access. If there are other journal repositories in our Library that have it, I've missed them. I'm stumped. If someone can help out, future readers will be better off...

I do not at all mind having to hunt down the article in the proper repository (keeps me from getting lazy, lol). Just point me in the right direction and off I'll go! Oona Wikiwalker (talk) 07:19, 27 February 2024 (UTC)

The cited articles are also available online for a (hefty) fee at GeoScienceWorld; see here and here. The fee imposed suggests that the publisher enforces its copyright.  --Lambiam 11:38, 27 February 2024 (UTC)

But Library access is within copyright restrictions because they pay for access for their patrons. That's what I'm inquiring after; did I miss a collection within our library that contains this journal?

Oona Wikiwalker (talk) 23:01, 27 February 2024 (UTC)

I've checked the Wikipedia Library and it doesn't provide access. I have added the doi and bibcode to the cite to make it a bit easier for readers to click to the journal. Mike Turnbull (talk) 16:48, 28 February 2024 (UTC)

It's long been an issue for those of us in Wikipedia:WikiProject Earthquakes, particularly for US earthquakes. You either pay the money or take a trip to a library that stocks that journal, which is generally easier said than done. Mikenorton (talk) 17:28, 28 February 2024 (UTC)

A mistake in our article Four momentum

The third paragraph states: p⁰ = E/c², with p⁰ being the momentum in the time dimension.

I'd fixed it: p⁰ = E/c, but somebody reverted. Would you like to give your opinion? HOTmag (talk) 18:06, 27 February 2024 (UTC)

The version with c² is formally correct. Another matter is whether that should be in the lede of the article (and who is "some authors"?). I don't think I've ever seen that convention — whenever I've seen $x^{0}=t$ it was due to setting $c=1$, in which case $p_{0}=E$. --Wrongfilter (talk) 18:13, 27 February 2024 (UTC)

I admit I'm quite surprised. The four momentum is (p⁰, p¹, p², p³), traditionally written also as (E/c, p_x, p_y, p_z), hence: p⁰ = E/c. On the other hand, both sides of the current expression (in the lede): p⁰ = E/c² don't have the same units, so how can it be correct? HOTmag (talk) 18:29, 27 February 2024 (UTC)

I wrote "formally correct", in some weird unit system. I don't think it makes much sense, which is why I removed it from the article. --Wrongfilter (talk) 18:38, 27 February 2024 (UTC)

Which one of the following three statements do you find "weird"?

1. The four momentum is (p⁰, p¹, p², p³).

2. The four momentum is (E/c, p_x, p_y, p_z).

3. Hence: p⁰ = E/c.

HOTmag (talk) 18:51, 27 February 2024 (UTC)

Oh please, give it a rest. The statement in the article had a context that you keep ignoring. --Wrongfilter (talk) 18:55, 27 February 2024 (UTC)

You are allowed to avoid answering my last question, which has nothing to do with the article. Actually, regardless of the article, I was quite curious to know whether you also considered the third statement p⁰ = E/c (which had been my correction before it was reverted): as "weird", while I assumed you accepted the two statements it followed, and that's why I asked you my last question, regardless of the article. But again, nobody forces you to answer my last question. HOTmag (talk) 19:44, 27 February 2024 (UTC)

$p^{0}=E/c$ is standard, nothing weird about that. --Wrongfilter (talk) 20:15, 27 February 2024 (UTC)

So (back to article), my correction p⁰ = E/c in the article was right, and it didn't have to be reverted to p⁰ = E/c². That's what I wanted to know from the very beginning: whether the user who reverted my correction was wrong (as I thought). HOTmag (talk) 22:26, 27 February 2024 (UTC)

No you were wrong in that context. That bit has been removed now from the article because what was there isn't common and as shown here will just cause confusion. It is best to have all the elements using the same units. NadVolum (talk) 23:25, 27 February 2024 (UTC)

I haven't asked about whether the context justified mentioning this formula there, nor about why the whole formula was eventually removed. I've only asked about whether, my replacing the original formulation p⁰ = E/c² by the correct formula p⁰ = E/c, had been a better version than the original version p⁰ = E/c². In other words, I've wanted to know if the user who reverted my correction p⁰ = E/c back to the original version p⁰ = E/c² made the article worse. I think they did, and I've wanted to get your opinion about what I think. All of that is quite clear in my first post, that had actually been posted before the whole formula was eventually removed. HOTmag (talk) 08:07, 28 February 2024 (UTC)

The original statement was a conditional, with an "if... then..." structure. The formula that you changed was in the "then..." part, and was correct under the condition stated in the "if..." part. By changing the "then..." part without taking into account the "if..." part, you introduced an error, and the other user was right in reverting your edit. --Wrongfilter (talk) 08:38, 28 February 2024 (UTC)

I got it now. Thank you. I understand now that the original "then" part (before I changed it) would be correct assuming that the "if" part were correct. However, the "if" part really introduces a definition I've never come across, and should be omitted. HOTmag (talk) 09:52, 28 February 2024 (UTC)

Parachutes and terminal velocity

Hi, I'm currently doing a lab using a small model parachute. As I currently understand it, the terminal velocity can be calculated by equating the weight of the object to the drag force, given by the drag equation. My question is, how can I use this terminal velocity to estimate the time it would take an object to fall a set distance? Do I assume the object starts out at terminal velocity? Do I use a SUVAT equation? Thanks! ARandomName123 (talk)^{Ping me!} 23:05, 27 February 2024 (UTC)

the time to accelerate to the terminal velocity needs to be accounted for. RudolfRed (talk) 00:22, 28 February 2024 (UTC)

So one needs to integrate the varying acceleration. Poor ARandomName123. Zarnivop (talk) 02:18, 28 February 2024 (UTC)

Integration is the bane of my existence. I guess I'll just approximate it then. Thanks for the help! ARandomName123 (talk)^{Ping me!} 03:09, 28 February 2024 (UTC)

Take the height of the fall, divide by the terminal velocity. That gives you a rough answer. So if the falling object gets close to terminal velocity after just a short fraction of the fall (for a human with a parachute, around 50 metres) and the fall is short compared to the scale height of the atmosphere (around 7 kilometres on Earth; else the changing density changes the terminal velocity) and you don't care too much about the second digit of your answer, this may be good enough. Otherwise, you have to integrate the equations of motion. PiusImpavidus (talk) 11:18, 28 February 2024 (UTC)

Yes, that's what I ended up doing. It wasn't actually too far off from the experimental result, only a few fractions of a second. Thanks! ARandomName123 (talk)^{Ping me!} 13:37, 28 February 2024 (UTC)

In the following, we only consider the motion of an idealized point particle along a vertical line. We use the variables $t$ for time, $s$ for the particle's position above $0$, $v={\frac {ds}{dt}}$ for its velocity and $a={\frac {dv}{dt}}$ for its acceleration. Furthermore, $m$ stands for its mass and $g$ for the acceleration of gravity.

The force exerted on the particle is the sum of the forces due to drag and to gravity:

$F=F_{\text{drag}}+F_{\text{grav}}=m\kappa v^{2}-mg=m(\kappa v^{2}-g),$

in which $F_{\text{drag}},$ whose form given by the drag equation, contains a yet unknown constant $\kappa .$ Its value may be determined if the terminal velocity $v_{\infty }$ is known, as follows. Terminal velocity is the limit velocity, approximated arbitrarily closely as the acceleration $a$ tends to $0,$ which means (since then $F=ma$ tends to 0) that the upward drag force cancels the downward gravitational force, or $\kappa v_{\infty }^{2}-g=0.$ It follows that $\kappa =g/v_{\infty }^{2},$ so now we have

${\frac {dv}{dt}}={\frac {F}{m}}=\kappa v^{2}-g=g\left({\frac {v^{2}}{v_{\infty }^{2}}}-1\right).$

This differential equation, combined with the condition that $v=0$ at time $t=0,$ the moment of release, is solved by:

$v=-v_{\infty }\tanh \left({\frac {g\,t}{v_{\infty }}}\right).$

Integrating again, with the condition that at time $t=0$ the particle is released from a given height $h,$ results in:

$s=h-{\frac {v_{\infty }^{2}}{g}}\ln \,\cosh \left({\frac {g\,t}{v_{\infty }}}\right).$

Solving $s=0$ for $t$ gives the time it takes the particle to reach ground level:

$t={\frac {v_{\infty }}{g}}\,{\text{arcosh}}\,\exp \left({\frac {g\,h}{v_{\infty }^{2}}}\right),$

in which ${\text{arcosh}}$ stands for the inverse hyperbolic cosine. Asymptotically, as $h$ tends to infinity, this is equal to

${\frac {h}{v_{\infty }}}+{\frac {v_{\infty }}{g}}\ln 2.$

 --Lambiam 00:36, 1 March 2024 (UTC)

I haven't learned inverse hyperbolic functions yet, but thanks for the detailed answer! ARandomName123 (talk)^{Ping me!} 02:38, 1 March 2024 (UTC)