Talk:Kinematics

"motion under gravity"

Removed the section on motion under gravity; it was poorly written, did not fit in with the rest of the article, and wrong. It seemed to say that Gallileo said that objects of different masses fall at different rates, which is clearly incorrect. 129.78.64.101 (talk) 14:33, 30 April 2008 (UTC)

+++ The material is copy-written by a textbook manufacture, nearly word for word. This article needs to be deleted and rewritten with out infringing on the published works+++++. —Preceding unsigned comment added by 72.135.111.140 (talk) 22:49, 23 February 2010 (UTC)

Sundry comments

This article contains lots of nonstandard notation. This is potentially confusing. More standard notation can be found in numerous physics texts including:

Halliday&Resnick, Johnson&Cutnell, Serway

This page contains several statements at the end about angular "position" and angular velocity that are non-general; that is, they only apply to a two dimensional or planar problem. MarcusMaximus 09:20, 25 August 2006 (UTC)

I added the section Algebraic equations to the Fundamental Equations heading. Give me your thoughts - I like that notation a bit better, it's more common, and a bit easier to understand. It may not reconcile with the vector notation used, but it is algebra. --Democritus the Minor 08:10, 6 September 2006 (UTC)

I notice that the change in the algebraic equations for linear kinematics are not notated consistently with the angular portion. I prefer describing linear kinematics using v_i and v_f instead of v and u because there is no obvious relationship between v and u in the equations, while the relationship between the velocities is vital in understanding the equations. Additionally, if initial and final angular displacements are used in describing angular kinematics, it seems reasonable to be consistent and use initial and final linear displacement to describe linear kinematics instead of using Δ. I believe it may be sufficient to simply notate the linear equations in both ways, which is what I have done.Tgilro01 20:59, 22 March 2007 (UTC)

Of course its confusing, its Physics! Anonymous 20:56, 14 May 2007 (UTC)

Physics is a beautiful subject which is not confusing in the least. Now as for this article it seems to be a crash course in kinematics and for that it is fine. —Preceding unsigned comment added by 117.194.225.190 (talk) 12:02, 30 March 2008 (UTC)

I removed the "First to Fourth Derivatives of Position" graphic. It was an image with four equations: f(x) = position, f′(x) = velocity, f′′(x) = acceleration, f′′′(x) = jerk, and f′′′′(x) = jounce. The term jerk is referenced in the text but not in the section containing the graphic. The term jounce is not referenced (though it is linked). The notation of the graphic was unexplained -- What is x? What is f(x)? What does the "prime" notation mean? r, dr/dt, d^2r/dt^2, etc would be more appropriate. Wouldn't these be better represented by a table? Is an article on kinematics even the appropriate place for this? --M0nstr42 (talk) 22:34, 2 June 2010 (UTC)

History and Explaination

While a purely mathematical treatment is nice, we should consider some plain language description of what each term does. We also need the history of kinematics, it's major developers, and the problems they were interested in solving.

Dhskep September 24, 2006 21:22 UTC

On Wikisource, you can now find The Kinematics of Machinery (1876) by Franz Reuleaux. The book is history in itself, but its Introduction also gives an overview of what happened from 1600 to 1876. I've linked all the persons mentioned to Wikipedia, and even wrote an article about José María Lanz who didn't have one. I guess Monge, Hachette and Lanz should be mentioned as pioneers here. --LA2 (talk) 12:27, 17 August 2010 (UTC)

Better late than never! Like Dhskep I have also written the same about history in the talk page of the equations of motion. I have skimmed through the book but didn't find any history about equations of motion. Everywhere in wikipedia, most articles of mathematics lack history.

Mathematics professionals seem to overlook the practical issues regarding brevity. At a time when publishing was expensive the writers and teachers needed to keep their work as brief as possible. Now with digitisation the primary hindrances have been overcome. Fed on a diet of barley some forget the taste of a multi-layered chocolate pastry overflowing with cream and icing :-) ;-) . I don't need to quote Pages 2-3, art. 0.1 Possible uses of history..., I. Grattan-Guinness, titled, From the Calculus to Set Theory, Princeton University Press.

Bkpsusmitaa (talk) 10:45, 14 August 2015 (UTC)

Relativistic kinematics?

Can anyone direct me to an article on relativistic kinematics (if one exists)? If there's no such article yet, it should probably be created either separately or as a subsection of kinematics. HEL 03:09, 5 November 2006 (UTC)

Specific notation confusion

I have long understood in Physics and Maths that difference (e.g. the difference between two speeds) is notated with a Greek 'Delta' Δ symbol.
In this article, it seems the notation is 'd' for difference. Whilst this makes sense, to people who have learnt it with the delta it is confusing.
Is this simply an Australian notation? Can we look into this some how? - Eps0n 05:24, 18 February 2007 (UTC)

Where do you see d being used for the difference? Δ is used for difference in the section "Algebraic equations." Elsewhere d is used only for the differential, as far as I can see. HEL 16:23, 18 February 2007 (UTC)

I agree, I have only found d to be used for the differential, which is its standard usage and is not the same as the difference described by delta. See [[1]]

notation

Why is u used to represent initial velocity? Is this standard notation? Jhausauer 02:14, 1 June 2007 (UTC)

v₀ or v_i are the standard methods of representing initial velocity.

The above comment is true however it must be added that u was formerly used for denoting initial velocity. The same applies the acceleration which can be denoted by either [i]f[/i] or by [i]a[/i] —Preceding unsigned comment added by 117.194.225.190 (talk) 12:05, 30 March 2008 (UTC)

The levels are not clearly defined and the terminology is questionable

This article seems a collage of statements taken from different chapters of a physics book, and assembled whithout a logical order, and without specifying clearly where they belong in. There are various levels of kinematics, and either they should be treated separately, starting from the simplest, or the simpler levels should be defined as special cases or the most complex level.

Level 1: Point (or point mass, or particle) motion In this context it is not correct, in my opinion, to use the expression "translational kinematics"; the word "translational" is used only for rigid body motion.

Sublevels:

• Linear kinematics in 1-D
• Linear kinematics in 2-D and 3-D
• Angular kinematics in 2-D and 3-D (for instance, in 3-D, the angular position of a particle rotating about a point cannot be described by a simple angle; you need a unit vector or an angular position vector...)

Level 2: Rigid body motion Here, and only here, you define the concepts of translation and rotation (see rigid body).

Sublevels:

• Linear kinematics in 2-D and 3-D, which describes translation (it is the same as above, but first you need to select a point fixed on the body)
• Angular kinematics in 2-D and 3-D, which describes the rotation (and this is a complex part, with several different conventions, explained in rotation (geometry))

Paolo.dL 22:06, 28 June 2007 (UTC)

Position vector: r or s?

Please do not use the symbol s for a position vector but only the symbol r. s is used in mathmatics and physics for 's = the distance along some curve C in space as measured from some fixed point' (cited from Standard Mathematical Tables). And 'ds = the vector element of a curve (=t ds)'(same source) It is extremely confusing to use s for a position vector. In it's whole this is a confusing text, mixing different elements together with not so clear relationships with each other.78.21.233.253 (talk) 16:54, 29 November 2007 (UTC) (I am Huibc in the Dutch Wikipedia)

Broken Links

It appears that links 1.3.1 to 1.3.4 are broken, that is, they don't move you to their corresponding subsections when clicked... —Preceding unsigned comment added by Jwedes (talk • contribs) 18:34, 25 August 2008 (UTC)

Symbols and their meanings

Most of the symbols here are quite ccomplex. In my first year at university studying Mech. Eng. this was even more complicated as you had to learn what each symbol stood for. Instead of writing down the equations can we have a table to the side telling us what each symbol means and how it is pronounced? NarSakSasLee (talk) 22:14, 23 August 2009 (UTC)

dynamics and cinematics

cinematics: "describes the motion of objects without consideration of the causes leading to the motion."

dynamics: "customarily refers to the time evolution of physical processes."

in the first one the authors leave out that the motion of objects appears over time. also a problem is that it´s arbitrary what a cause is, because one can see the "cause" as a part of the system and this system has been caused by some other causes. one could also zoom in and say that the moving object can be divided into a caus and an effect. fact is that one has to blend out the affected object and blends out the cause while watching the affected object. it´s just a game of zooming in and out and arbitrarily either blending out the surroundings or the details as if they would disappear.

91.51.204.80 (talk) 14:25, 2 September 2009 (UTC)

"Kinematics"

Have changed the initial sections today. The exposition wasn't smooth. There was no mention of the idea of rest and motion stemming out from the fundamental frame dependence of position vectors. Come to think of it, there was no mention of reference frames at all. Have added that. Will get to the sections after "Integral relations" soon. The rest of it needs much standardization. "Linear Motion" shouldn't be the primary heading. It should be "Particle Kinematics" and "Linear Motion" Should appear as a sub-category in Particle Kinematics. Further, the average quantities were introduced after the instantaneous ones. Scanty remarks were made for the limiting definition of velocity or acceleration. Soumengoswami10 (talk) 19:27, 18 November 2009 (UTC)soumengoswami10

can't begin an article with a quote

• can't begin an article with a quote. no can do. where should it be moved? – Peacock.Lane 10:46, 28 February 2011 (UTC)

Revisions to Kinematics

This article focusses on particle kinematics which is essentially the differential geometry of a curves. I would like to stream-line this presentation of particle kinematics somewhat, and then expand it to the kinematics of rigid bodies. Things get more interesting when we consider the kinematics of articulated systems such as in robot kinematics. I will try to make that connection. Prof McCarthy (talk) 13:11, 25 December 2011 (UTC)

After revising the lead and adding section on rigid transformations, I am now examining the section on particle kinematics. This section seems to be repeating basic results of vector calculus. I am not sure I can salvage much of it. Prof McCarthy (talk) 07:19, 30 December 2011 (UTC)

I concluded that I could not fix the section on particle kinematics without deleting large portions, which did not seem inappropriate. So instead, I have appended a section on the position, velocity and acceleration of points in a moving rigid body. It is a rather detailed derivation that seems to balance the elementary nature of the section on particle kinematics. Maybe this will work. Prof McCarthy (talk) 04:44, 31 December 2011 (UTC)

Particle kinematics needs work

I think the section on particle kinematics appears to be general discussion of elementary physics. It discusses reference frames, the definition of the derivative, integration of ballistic trajectories, relative position vectors, and more. I will leave it alone, because I just cannot quite figure it out. Instead, I propose to work on the section on kinematic constraints, which is important and neglected. Prof McCarthy (talk) 07:33, 31 December 2011 (UTC)

Let me just add that the first sentence "Particle kinematics is the study of the kinematics of a single particle." captures the redundancy that is present in much of this section. Having said this, it is not correct that there is nothing that is useful. It is just difficult to sort it out. Prof McCarthy (talk) 07:39, 31 December 2011 (UTC)

Coordinate systems

I think the section Kinematics#Coordinate systems can be revised so the formulas for a moving particle are easier to understand. I will work on this on my user page and drop it in for evaluation. Prof McCarthy (talk) 21:46, 31 December 2011 (UTC)

Point object in circular motion

This section seems to be a duplicate of the position, velocity and acceleration in cylindrical coordinates. I will add it as a special case. Prof McCarthy (talk) 22:50, 31 December 2011 (UTC)

Commented out sections

I have commented out a few sections because they were non-standard, duplicative or irrelevant. I think this helps the remaining material flow much better. Prof McCarthy (talk) 23:28, 31 December 2011 (UTC)

Picture for the introduction

Watt's steam engine

The Boulton & Watt Steam Engine, 1784.

I have to say that Paolo.dl's edits to this article are welcome. Unfortunately, he puts his long explanations into the edit description, which is interesting reading but makes it difficult to respond. He deleted the picture of the steam engine even though almost every feature illustrates an aspect of the study of kinematics. I believe he is well-meaning and would like to see something more mathematical in appearance, but prefers a blank page until someone else finds something to his liking. This means that he does not see (i) the straight line movement in the expansion cylinder guided by Watt's straight-line linkage, (ii) the four-bar linkage formed by the walking beam input crank and its coupler that drives the output crank rotating the main drive pulley, and (iii) the fly-ball governor which is rotating slider crank that provides feedback control of the power source. These are all fundamental kinematic elements assembled into arguably one of the most important technological developments in history, the Boulton-Watt steam engine. It is probably fair to say that the ability to generate mechanical forces and movement with a steam engine also drove the study of mechanics, in general, and kinematics, in particular. Prof McCarthy (talk) 16:50, 4 January 2012 (UTC)

Please, do not try to guess what I can see or not see in the picture of the steam engine (especially if you assume that I can't see). I greatly appreciate most of your edits, so I value your opinion, but I removed the picture because I believe that the kinematics of an engine is not an appropriate example in the introduction of an article about kinematics. Moreover, that picture was not even about the kinematics of an engine, but showed only its structure (the motion of the engine is not described at all; no kinematic quantity is mentioned or represented). The picture belongs in the introductions of articles such as steam engine or mechanical system.

In physics textbooks, kinematics is first described for the motion of a point, then for rigid bodies, and sometimes also for multi-body systems. The motion of a mechanical system, therefore, represents one of the most complex possible examples of motion. Astronomy, mechanical engineering, biomechanics, are all sciences in which kinematics is applied to study celestial bodies, mechanical systems, multi-link biological systems. But first of all, we need to explain how (linear) kinematics describes the motion of a point. We need to assume that the reader knows nothing about kinematics. What would you say if a biomechanist inserted in the introduction a picture about the kinematics of a twisting somersault? For a biomechanist, that would be much more attractive than the steam engine. And what about rotation, revolution, precession and nutation of Earth? These are fascinating examples, however they are complex, and our readers are neither advanced students, nor mechanical engineers, nor biomechanists. They need much simpler examples. I know that this may be boring for expert editors like you (according to Hawking, Albert Einstein was offered positions in many Universities, but he chose a "research position with the Prussian Academy of Sciences in Berlin because it freed him from teaching duties"), but our main purpose is not having fun. We are here to be of service for our readers, and most of them are beginners.

Last but not least, the kinematics of mechanical systems is not described in this article (and this is consistent with most introductory physics texbooks, which focus mainly on the kinematics of particles and rigid bodies). According to WP:MOS, the introduction of an article should only summarize the article's main topics. The picture about the steam engine is not appropriate for any of the sections of the article, and therefore not appropriate for its introduction.

Paolo.dL (talk) 18:07, 4 January 2012 (UTC)

Needless to say, I disagree with most of our colleague's statements. I hope I was clear in the discussion above that the steam engine is an excellent portrayal of kinematics. I can also say that I would very much appreciate a picture about the kinematics of a twisting somersault, and one about the rotation, revolution, precession and nutation of the earth. But, I see that even these are considered too complex for our readers and not helpful for beginners. Again, I disagree. Finally, among the very first problems in physics are the kinematics of particles moving along a straight line and moving on a circle. In my opinion, even beginners would recognize that the components of a steam engine contain many particles that are moving along straight lines and on various circles. Prof McCarthy (talk) 06:48, 5 January 2012 (UTC)

"Needless to say, I disagree"? Are you assuming that everybody should agree with you? Discussions imply the need to interact. On Wikipedia, consensus is needed to take decisions. I will expand on just one of the points you ignored in your reply: the article is not about objects which move. Showing an object which can move (or even an animation of a moving object) is not enough to illustrate kinematics. The picture should at least show a trajectory, possibly an "arrow" representing a kinematic quantity. In other words, something that "describes" the motion.

Since you totally rejected my points, I suggest to wait for contributions by other editors to this discussion. Paolo.dL (talk) 12:32, 5 January 2012 (UTC)

I believe we have established that there is no consensus regarding your criteria for acceptable figures for this article. Therefore, please replace it until you can find another editor who agrees with you. Prof McCarthy (talk) 15:20, 5 January 2012 (UTC)

Your reply appears so naively biased to me that I feel empahtetic, rather than offended. You should either have written: "I have established that there is no consensus regarding your criteria" (which would have revealed more openly your bias) or "we have established that there is no consensus between us regarding this topic" (which is true). I remember what I felt whenever somebody reverted some edits of mine without my approval, in articles that I highly contributed to improve, sometimes even without offering adeguate edit summaries. So, I understand your passion, but still you need to abide to the policies on Wikipedia: no editor can impose his own criteria, not even one who completely wrote an ecellent article from scratch.

In this case, accoding to good sense, politeness, respect for each other and Wikipedia policies, we need to wait for contributions by other editors, and in the meantime we need to accept the situation as it was before our edits (not just before my edits). This means that the introduction should stay without a figure, because the figure I removed was added by you on the 29th of December with this edit. Paolo.dL (talk) 16:52, 5 January 2012 (UTC)

Wheel

Figure from Kinematics of Machinery 1876: A wheel, constructed of perfectly rigid material, supported between two mounts.

This figure has an arrow and comes from a book called Kinematics of Machinery. Prof McCarthy (talk) 00:28, 6 January 2012 (UTC)

Yet another figure showing a machine. Nothing in it helps the reader to understand what kinematic is (there is a vector Q which is not described, whose meaning is quite difficult to understand even for me; it may possibly be meant to describe an angular kinematic quantity such as angular velocity or angular acceleration, but its direction is not consistent with the right hand rule, and its position in the picture is quite unusual for a such a vector). Please do not insert new figures in the introduction of the article. Please patiently wait for the opinion of other editors in this talk page. Paolo.dL (talk) 10:21, 6 January 2012 (UTC)

Four-bar linkage

Illustration of a four-bar linkage from Kinematics of Machinery, 1876

Well, it seems I am in the position of trying to find a figure that satisfies the personal tastes of our colleague Paolo.dL. As far as I can tell he does not want illustrations of the moving parts of a complex machine like the Boulton and Watt steam engine, nor does he want the simplest example of rotating disk which he considers to be a machine. He does want arrows and some mathematical display of movement. I am fairly certain, therefore, that he will not accept the four-bar linkage illustration, though it is the focus of most courses on Kinematics. I am at a loss. Prof McCarthy (talk) 16:59, 6 January 2012 (UTC)

[NOTE: I removed the new section title "Paolo.dL and figures" that Prof McCarthy just inserted, and substituted it with a new sub-section title: "Four-bar linkage". A new section title would be highly misleading, as the discussion topic did not change at all (see Wikipedia:Refactoring talk pages for guidelines). Moreover, there cannot be a discussion about "Paolo.dL and figures". We need to focus on the article, not on the opinion of a single editor (see also WP:NPA). This is a discussion about figures in the introduction, not about me, McCarty, or me versus McCarthy. Contributors to this discussion will give their opinion freely on this topic, and not only about the comments of a single editor. I also changed the section title from "Paolo dL's edits" to "Picture for the introduction" (the picture was inserted by McCarthy and removed by me, but we should discuss and have been discussing about the picture, not about who inserted or removed it). This will hopefully attract the attention of other contributors. Paolo.dL (talk) 20:30, 6 January 2012 (UTC)]

About this new figure that McCarthy proposes, it is very similar to the previous ones. Also, it seems he confuses kinematics with kinematics of machines. His sentence "the four-bar linkage illustration [...] is the focus of most courses on Kinematics" reveals his bias, although I am sure he perfectly knows that kinematics is a branch of physics, not a branch of mechanical engineering. Kinematics was first used by Galileo and Newton to describe the motion of projectiles, sliding objects, balls rolling along inclined planes, penduli, and celestial bodies. As far as I know, most physicists do not describe the motion of machines when they explain kinematics in undergraduate courses.

McCarthy has been totally ignoring most of my comments, and this unfortunately prevents a constructive discussion between us. Possibly, he thinks exactly the same thing about me. This is why I insist that we need the opinion of other editors, and keep asking him to be patient. Paolo.dL (talk) 20:30, 6 January 2012 (UTC)

Ball rolling or block sliding along inclined plane

As, I wrote above, kinematics was first used to describe the motion of projectiles, sliding objects, balls rolling along inclined planes, penduli, and celestial bodies. A description of Galileo's experiments with sliding blocks or rolling balls, for instance, is perfect to introduce the concept of kinematics. For instance, see this intuitively appealing interactive animation, where space, time, and time-variant values of velocity and acceleration are shown. (Remember to press the start button; if you can't see the animation, probably you need to install Java runtime environment). By the way, a sliding plane is considered to be a simple machine, but the focus here is on the kinematics of the block (which is not a machine). Paolo.dL (talk) 21:11, 6 January 2012 (UTC)

Reformatting Talk Pages

I believe it is inappropriate to reformat talk pages, particular entries by other editors. This removes the historical nature of an evolving conversation. By taking this action Paolo.dL has reached beyond his assumption of final authority regarding my choice of figure, to domination of my responses to his actions. This behavior should be banned on Wikipedia. The fact that nothing will happen should put to rest the myth that there is another editor who will regulate his actions. Prof McCarthy (talk) 22:54, 6 January 2012 (UTC)

Refactoring talk pages is allowed in Wikipedia. Especially if it is mild and respectfully performed as in this case. I only changed two section titles, which referred directly and inappropriately to me rather than to the real topic of the discussion, and added two other subsection titles. All the new titles ("Picture for the introduction", "Watt's steam engine", "Wheel", and "Four-bar linkage") showed no point of view and objectively described the contents of the following text, respecting the "historical nature" of the discussion. See also my note above, where I explained everything. This kind of mild refactoring can significantly improve the readability of the page, and this, according to WP:Refactor, is appropriate and in some cases desirable. See also the last comment by Johnuniq on the talk page of WP:Refactor, about "Changing section titles". Johnuniq wrote: "I did one such change recently because the article talk page section heading was addressed to a specific editor, and it is usually best to avoid unnecessary confrontation". Paolo.dL (talk) 00:30, 7 January 2012 (UTC)

Robotics attention needed

• Read through
• Assess
• Add notes for any further work needed

Chaosdruid (talk) 11:27, 28 February 2012 (UTC)

Revisions

I am sorry for the long chain of revisions. I just wanted to adjust the order of topics so that it proceeded from particle kinematics to planar rigid body kinematics and then to spatial rigid body kinematics, but the notation and concepts needed sorting out much more than I expected. Prof McCarthy (talk) 10:03, 22 June 2012 (UTC)

Symbol s not introduced

In the section "Velocity and speed" the symbol 's' (distance travelled) is used a few times, but it has never been introduced/defined. — Preceding unsigned comment added by Leen Droogendijk (talk • contribs) 13:08, 26 June 2012 (UTC)

I will make this correction. Thank you Prof McCarthy (talk) 04:58, 27 June 2012 (UTC)

Revisions to section on the rotation of a rigid body

I am sorry that I have made a long list of revisions without explanation. I thought I could do this easily and quickly, but it did not go as smoothly as I had hoped. I have provided the matrix calculations that define angular velocity and angular acceleration for a rotating body. Prof McCarthy (talk) 16:38, 13 July 2012 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Kinematics/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Adopted by Willow from the Physics WikiProject.

Last edited at 18:22, 3 October 2006 (UTC). Substituted at 21:14, 29 April 2016 (UTC)

Invalid division by a vector in "Particle trajectories under constant acceleration"

The substitution should take into account $\mathbf {A} t$ instead of only $t$:

$\mathbf {A} t=\mathbf {V} -\mathbf {V} _{0}$

$\mathbf {P} -\mathbf {P} _{0}=\left({\frac {\mathbf {V} +\mathbf {V} _{0}}{2}}\right)t$

$\left(\mathbf {P} -\mathbf {P} _{0}\right)\circ \mathbf {A} =\left({\frac {\mathbf {V} +\mathbf {V} _{0}}{2}}\right)\circ \mathbf {A} t$

$(\mathbf {P} -\mathbf {P} _{0})\circ \mathbf {A} =\left(\mathbf {V} -\mathbf {V} _{0}\right)\circ \left({\frac {\mathbf {V} +\mathbf {V} _{0}}{2}}\right)\ ,$

— Preceding unsigned comment added by Nooser (talk • contribs) 13:52, 24 September 2016 (UTC)

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Copying section on "Kinematics of a particle trajectory in a non-rotating frame of reference" to Wikiversity

I am copying this section over to a course on robotic mechanics on Wikiversity. https://en.wikiversity.org/wiki/Robotic_Mechanics_and_Modeling/Kinematics Admazzeo (talk) 20:13, 20 January 2020 (UTC)

Relative Acceleration problems

This section is unfocussed, lacks formal vector algebra to match earlier sections, and poorly explained equation derivation for constant acceleration (can be shown from time dependent acceleration in few steps).

V(t) integration assumes constant acceleration without mention.

'translation' not 'displacement'

At 1st citation needed: Strike [c. n.]. Replace phrase with "By definition of the dot product" before equation. There is no need to cite a fundamental vector algebra definition.

Strike "In the case of acceleration..." Replace with "For the case of linear motion, the difference of positions vector, velocity vectors, and acceleration vectors will be colinear. All angles between vectors will be 0° or 180° giving 1 or -1 for the cosine of these angles respectively. The vectors can be written as their scalar magnitudes:

v^2 = v0^2 + 2•a•(x - x0) = v0^2 + 2•a•∆x

Strike following sentence. Define scalar variables after next equation, "where v = |V|, a = |A|, and ∆x = |P| - |P0| = x - x0."

'Curvelinear', not 'curvaceous.'

Remainder of section from "We also know that..." repeats the earlier derivation, is circuitous, and poorly annotated. If there is no WP page that diagrams the time differentials of position, velocity, and acceleration and explains their relations anywhere else, it could be useful here. The velocity integration graphical representation and derivation is awkward and unnecessarily weedy. The graph is lacking significant annotations and those that are there are unhelpful. (If I had to, I would have used the area of a trapezoid formula instead of calculating areas of a rectangle and triangle and then adding them together.) The text here uses a lot of unintelligible and incorrect terminology. The graph caption is a good example. Figure 2 is out of place here having undefined terms and parameters involving non-inertial reference frames yet to be addressed. EvolutionOfTruth (talk) 02:15, 31 August 2020 (UTC)

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