Is velocity ever a scalar quantity?

[Charles Link]

It may be of interest to the OP that English uses the word velocity to indicate it being a vector and the word speed when referring to the magnitude of the velocity vector. From what I have read, the German language doesn't have the distinction of these two and uses the word Geschwindigkeit for both velocity and speed. They then need to specify whether they are referring to the vector, or its magnitude. In any case, I thought the inputs above were very good, and did a good job explaining how velocity, except when used very loosely, refers to a vector.

Edit: @fresh_42 You have German as your native language= Gibt es nur das einzige Wort Geschwindigkeit? (Is there only the one word Geschwindigkeit?)

 

[Orodruin]

It may be of interest to the OP that English uses the word velocity to indicate it being a vector and the word speed when referring to the magnitude of the velocity vector. From what I have read, the German language doesn't have the distinction of these two and uses the word Geschwindigkeit for both velocity and speed. They then need to specify whether they are referring to the vector, or its magnitude. In any case, I thought the inputs above were very good, and did a good job explaining how velocity, except when used very loosely, refers to a vector.

Interesting. In Swedish we have distinct words: hastighet and fart. Usually words are pretty 1-1-mapped to German …

(We also have a saying that translates funnily to Swenglish:
Det är inte farten som dödar, det är smällen. -> It is not the fart that kills, it is the smäll.
”Smäll” being pronounced as ”smell” and being Swedish for ”impact”)

 

[blue raven]

To understand what velocity is, would be quite useful a diagram, such as the one below. By the way, speed is not the magnitude of velocity.

A car starts to move with a constant speed from point A to point B, passing through the intermediate point C on this gray road. Velocity is defined as a vector, $\vec{v}$ , which means it has one direction and one magnitude. Velocity is calculated as displacement (blue vector) $\vec {d}_{AB}$ divided by time, regardless of the length of the path ACB.
$$\vec{v}=\dfrac{\vec{d}_{AB}}{t}$$
This is the magnitude of the vector velocity, and it is not the speed of the car.
Speed is the one defined (or achieved) on the curved path ACB. A car can accelerate from A to C, and its speed is calculated using this distance, using suvat. You cannot calculate the magnitude of the vector velocity using suvat because the displacement AB is different from the path ACB. The only possibility to have speed=velocity would be a straight gray path from A to B. And by the way, a line that has one dimension 1D (just length), do not imply that this line is not curved in 2D or 3D.
So it seems that Khan is right. You can calculate the magnitude of the vector velocity $\vec{v}$ only if you know the magnitude of the displacement vector $\vec{d}$ .

 

[Charles Link]

I disagree completely=the velocity vector changes direction as the car goes through this curved path, but I think even you probably know that.

 

[blue raven]

And the direction of vector velocity $\vec{v}$ is given by what? By vector displacement $\vec{d}$. You can find the vector displacement only when you choose two points A and B on the curved gray path. So, this is exactly what I did, I have chosen two points on the gray path. When the car travels from A to B, this is the vector velocity, it has one direction and one magnitude, not two, not three.

 

[Orodruin]

And the direction of vector velocity $\vec{v}$ is given by what? By vector displacement $\vec{d}$. You can find the vector displacement only when you choose two points A and B on the curved gray path. So, this is exactly what I did, I have chosen two points on the gray path. When the car travels from A to B, this is the vector velocity, it has one direction and one magnitude, not two, not three.

You are thinking of average velocity over some finite time. This is not how instantaneous velocity is defined - it is defined as a limit when the time interval goes to zero. Speed is defined as the magnitude of instantaneous velocity. Simple as that.

Claiming something else is just wrong.

 

[robphy]

The velocity vector at a given time is a tangent vector to the trajectory (which gotten by taking the limit mentioned by @Orodruin ).

 

[blue raven]

You are thinking of average velocity over some finite time. This is not how instantaneous velocity is defined - it is defined as a limit when the time interval goes to zero. Speed is defined as the magnitude of instantaneous velocity. Simple as that.

Claiming something else is just wrong.

So Encyclopedia Britannica is wrong?
"Speed is the time rate at which an object is moving along a path, while velocity is the rate and direction of an object’s movement. Put another way, speed is a scalar value, while velocity is a vector. For example, 50 km/hr (31 mph) describes the speed at which a car is traveling along a road, while 50 km/hr west describes the velocity at which it is traveling."
https://www.britannica.com/story/whats-the-difference-between-speed-and-velocity

NASA about instantaneous velocity, does not say anything about any speed. Please provide a credible link for your allegation.
The velocity -V of the object through the domain is the change of the location with respect to time. In the X - direction, the average velocity is the displacement divided by the time interval:
V = (x1 - x0) / (t1 - t0)
This is just an average velocity and the object might speed up and slow down between points "0" and "1". At any instant, the object could have a velocity that is different than the average. If we shrink the time difference down to a very small (differential) size, we can define the instantaneous velocity to be the differential change in position divided by the differential change in time;
V = dx / dt
https://www.grc.nasa.gov/www/k-12/airplane/disvelac.html
Velocity vs speed
https://www.physicsclassroom.com/class/1dkin/lesson-1/speed-and-velocity

 

[weirdoguy]

So Encyclopedia Britannica is wrong?

Encylopedia Britannica contradicts what you said. Read your sources. You are wrong, according to them!

Please provide a credible link for your allegation.

Every textbook on basic physics that uses calculus. Please mind you, a lot of us are physicists, we kind of know what we are talking about. I don't like arguments from authority, but we are talking about one of the most basic things in physics, and you seem to be a little too argumentative.

 

[blue raven]

The velocity vector at a given time is a tangent vector to the trajectory (which gotten by taking the limit mentioned by @Orodruin ).

This is correct, but again a diagram would be quite useful to understand the difference between average velocity versus instantaneous velocity, such as the one below.

The vector velocity $\vec{v}$ is given by the vector displacement $\vec{d}$. When we reduce the distance between the two points, this vector comes closer to the apex of the curve. We move the displacement from AB to CD to EF, and then the two last points GH are very close. Now the last displacement vector $\vec{d}_{4}$ is tangent to the path indeed, while it becomes very small. So this is the explanation.

 

[Orodruin]

This is correct, but again a diagram would be quite useful to understand the difference between average velocity versus instantaneous velocity, such as the one below.
View attachment 355625

The vector velocity $\vec{v}$ is given by the vector displacement $\vec{d}$. When we reduce the distance between the two points, this vector comes closer to the apex of the curve. We move the displacement from AB to CD to EF, and then the two last points GH are very close. Now the last displacement vector $\vec{d}_{4}$ is tangent to the path indeed, while it becomes very small. So this is the explanation.

The displacement becomes small, but so does the time. These two effects will cancel ouf to give a finite velocity.

You also do not need to approach the apex. The velocity along your curve will be different depending on the position you consider.

Please provide a credible link for your allegation.

Any basic textbook on classical mechanics will do. Try Landau-Lifshitz or the Feynman lectures on physics for example. There really is no way of saying this but you are simply wrong and any introductory textbook will tell you so.

 

[blue raven]

You are thinking of average velocity over some finite time. This is not how instantaneous velocity is defined - it is defined as a limit when the time interval goes to zero. Speed is defined as the magnitude of instantaneous velocity. Simple as that.

Claiming something else is just wrong.

In the image below are depicted both, average velocity, by displacement $\vec{d_{1}}$ , and also instant velocity provided by displacement $\vec{d_{4}}$ .

You say that speed is the magnitude of velocity, which is correct only for instantaneous velocity, I get it, when the displacement from point G to point H is very small and is the same as the trajectory from G to H, which I explicitly explained in my first post that velocity = speed only when the trajectory is straight, which is the case here.
When we are referring to average velocity, speed is not velocity, because the curved path ACB is not the straight displacement AB. The English language provides this distinction between speed (scalar) and velocity (vector), so why don't you use it? By insisting on speaking about speed = velocity, you can only cause confusion, because then you have to explain further that this is only valid for instantaneous velocity. Velocity has a magnitude, but it is not speed, not in English.

In everyday usage, the terms “speed” and “velocity” are used interchangeably. In physics, however, they are distinct quantities. Speed is a scalar quantity and has only magnitude. Velocity, on the other hand, is a vector quantity and so has both magnitude and direction. This distinction becomes more apparent when we calculate average speed and velocity.

To illustrate the difference between average speed and average velocity, consider the following additional example. Imagine you are walking in a small rectangle. You walk three meters north, four meters east, three meters south, and another four meters west. The entire walk takes you 30 seconds. If you are calculating average speed, you would calculate the entire distance (3 + 4 + 3 + 4 = 14 meters) over the total time, 30 seconds. From this, you would get an average speed of 14/30 = 0.47 m/s. When calculating average velocity, however, you are looking at the displacement over time. Because you walked in a full rectangle and ended up exactly where you started, your displacement is 0 meters. Therefore, your average velocity, or displacement over time, would be 0 m/s.
https://phys.libretexts.org/Bookshelves/University_Physics/Physics_(Boundless)/2:_Kinematics/2.2:_Speed_and_Velocity

Landau here, feel free to point out whatever you may consider necessary
https://archive.org/details/landau-and-lifshitz-physics-textbooks-series/Vol 1 - Landau, Lifshitz - Mechanics (3rd ed, 1976)/page/1/mode/2up?view=theater

 

[berkeman]

Thread closed for Moderation...

 

[berkeman]

@blue raven has been thread banned from replying in this thread because of misinformation, and the thread is reopened provisionally.

 

[Orodruin]

explained in my first post that velocity = speed only when the trajectory is straight, which is the case here.

This is wrong. Velocity is not equal to speed. It never happens because velocity is a vector and speed is a scalar - they cannot be equal to each other. Speed is defined as the magnitude of velocity.

When velocity is mentioned without any other qualifiers, what is being meant is the instantaneous velocity, not the average velocity.

Your path example is never straight as it has non-zero curvature everywhere. That you can approximate it by a straight line whose direction is the direction of velocity is s separate issue. Regardless, speed is the magnitude of velocity along the entire path.

 

[silenteuphony]

TL;DR Summary: Is the v in the suvat equations a scalar or a vector?

I thought velocity was always a vector quantity, one with both magnitude and direction.

When it came to the suvat equations, where v = final velocity, and u = initial velocity, I thought both of those were vector quantities, e.g;

v (final velocity) 112km/hr North

u (initial velocity) 0km/hr (I'm now asking; what do we put for direction when the object is initially stationary?)

But in a Khan Academy question they ask what does the letter v (lower case with no arrow above it, or anything else) stand for, and whether it’s a vector or a scalar.

I answered ‘velocity’ (it was multiple choice with no option for ‘final velocity’) and that it was a vector.

Their answer was;

“The symbol v represents speed, a scalar.”

I know speed is a scalar, but thought v stood for final velocity.

TL;DR Summary: Is the v in the suvat equations a scalar or a vector?

I thought velocity was always a vector quantity, one with both magnitude and direction.

When it came to the suvat equations, where v = final velocity, and u = initial velocity, I thought both of those were vector quantities, e.g;

v (final velocity) 112km/hr North

u (initial velocity) 0km/hr (I'm now asking; what do we put for direction when the object is initially stationary?)

But in a Khan Academy question they ask what does the letter v (lower case with no arrow above it, or anything else) stand for, and whether it’s a vector or a scalar.

I answered ‘velocity’ (it was multiple choice with no option for ‘final velocity’) and that it was a vector.

Their answer was;

“The symbol v represents speed, a scalar.”

I know speed is a scalar, but thought v stood for final velocity.

Strictly speaking, velocity is a vector, of course, but context is everything. What kind of problems were in the section they covered just before the question? If the motion in those problems was limited to one dimension, and the v was formatted like a regular scalar variable (i.e. the v wasn't in boldface, and didn't have an arrow over it), then it could definitely be interpreted as a regular scalar variable. Even if that was the case, however, it was still a bad answer, because they should be teaching definitions that are as standard and universal as possible, so that the knowledge you're learning has maximum transferability.

 

[blue raven]

I have posted on this topic after seeing a not very correct definition of speed by @Orodruin, which in the post #9 says “Speed is - by definition - the magnitude of velocity”. The point is that such chopped/shortened definition can be made only in the case of a motion in 1D, so I tried to underline this particularity. After that @Orodruin corrected its statement in post #41 saying that “Speed is defined as the magnitude of instantaneous velocity. Claiming something else is just wrong.” Indeed, this is the correct definition that can be found in textbooks, now @Orodruin is 100% right, and I tried to underline once again that when we are referring to speed, we should strictly stick to the “instantaneous velocity”, because in a motion in 2D or 3D any other velocity, such average velocity, has a magnitude different from speed. So, after I said that “speed is magnitude of velocity, is correct only for instantaneous velocity”, @berkeman decided that I “promoting content that is false or misguided”, banning me, see the screenshot

Now let me explain. In the case of 1D motion, it is correct to say that “speed is the magnitude of velocity”, because the direction of vector velocity cannot be changed (only reversed), so its magnitude cannot differ from speed.

In the case of a motion in 2D or 3D things change, and this chopped sentence above of speed is no longer valid. An object can move on a curved 2D-3D, trajectory, such as a projectile launched into the air, and now we can calculate the magnitude of any velocity vector we want, pointed in any direction of the space. Mathematically, the speed of the projectile along its curved trajectory is not the magnitude of its velocity along the Ox axis and is not the magnitude of its velocity along the Oy axis and is not the magnitude of its velocity arbitrarily oriented to some point in space, for example its velocity to the north. So, the only case in which “speed is magnitude of velocity”, is the case of INSTANTANEOUS VELOCITY. Now let’s look into the textbooks to see if it is true

Sears and Zemansky “University Physics”, 12 th edition, Chapter 3 entitled “Motion in two or three dimensions”, page 72,

The above excerpt from a physics manual, refers to “INSTANTANEOUS VELOCITY”. And apart from this definition which refers to “instantaneous velocity” there is no other definition of speed.

Halliday and Resnik, “Fundamentals of Physics”, 10 th, Chapter 2.2, page 18, the same statement

The definition above refers strictly to INSTANTANEOUS VELOCITY and there is no other definition of speed, this is the only one. So it is what I said, but using other words, trying to emphasize that we cannot use this definition for any other type of velocity.

So, when I said “Speed is the magnitude of velocity, only in the case of instantaneous velocity”, what do you mean @berkeman, that “I stubbornly promoted misinformation”? Apart from "instantaneous speed", is it possible that you may know of another type of velocity that can be used in this definition which is not presented in these manuals? So, can you write down @berkeman a single other velocity that can be used instead of "instantaneous velocity" in such definition of speed, which textbooks do not talk about? Please enlighten us! You can always plead the fifth amendment avoiding incriminating yourself, because if you know something different from what is written in the manuals, this is forbidden by the forum rules.

 

[PeroK]

In these introductory texts, the authors have emphasised the word "instantaneous" to ensure there is no confusion with average velocity. In physics the term velocity means instantaneous velocity. It never means average velocity.

In fact, this is true of a whole range of quantities: velocity, acceleration, momentum, force, etc. These do not need to be qualified by the word "instantaneous". Take Newton's second law:
$$\vec F = m\vec a$$That says "force equals mass times acceleration". No one says "instantaneous force equals mass times instantaneous acceleration". In fact, technically, it is explicity a differential equation:
$$\vec F = m\frac{d^2\vec r}{dt^2}$$Where $\vec r$ is the particle's displacement.

 

[Orodruin]

Now let me explain. In the case of 1D motion, it is correct to say that “speed is the magnitude of velocity”, because the direction of vector velocity cannot be changed (only reversed), so its magnitude cannot differ from speed.

This is wrong if you by ”velocity” do not mean instantaneous velocity. Consider the case of an object going from A to B and back. When the object comes back to A the average velocity is zero but the average speed is not.

As @PeroK has already mentioned regarding those texts, they are extremely introductory and only emphasize instantaneous for extra clarity. Average velocity on the other hand is always going to be called just that, never just ”velocity”.

 

[Orodruin]

Oh, and even more to the point here. The velocities entering into the SUVAT equations, ie, the subject of this thread, are the initial and final velocities. Those velocities are definitely instantaneous quantities so starting to talk about average velocity is simply going off on a tangent.

 

[pines-demon]

It may be of interest to the OP that English uses the word velocity to indicate it being a vector and the word speed when referring to the magnitude of the velocity vector. From what I have read, the German language doesn't have the distinction of these two and uses the word Geschwindigkeit for both velocity and speed. They then need to specify whether they are referring to the vector, or its magnitude. In any case, I thought the inputs above were very good, and did a good job explaining how velocity, except when used very loosely, refers to a vector.

It is the same in French, you just use vitesse for both if you need the vector you can specify by saying vecteur vitesse (velocity/speed vector) [some people like to use célérité for speed, which is the origin of using $c$ for the speed of light, but is not the norm and it is never used in advanced courses unless talking about waves]. Weirdly enough, in other closely-related languages like Spanish, you have two words again: rapidez (speed) and velocidad (velocity).

 

[Ibix]

some people like to use célérité for speed, which is the origin of using $c$ for the speed of light

Celerity also exists in English, as far as I know only for the (deservedly obscure) quantity $dx/d\tau$, which is the Lorentz gamma factor times the velocity. I don't think I've seen it used except for people dutifully mentioning its existence before going on to never use it again.

Weirdly enough, in other closely-related languages like Spanish, you have two words again: rapidez (speed) and velocidad (velocity).

But I don't think it has the same clear distinction as in English. For example, radar speed trap areas say "Control de velocidad por radar", not rapidez. And in physics I believe you always use velocidad, and would say velocidad vectorial or velocidad magnitud if you need to make the distinction (or so I was told, but by a chemist). I think rapidez is for less quantitative situations. Perhaps @mcastillo356 could comment.

Side note: in Galicia official signs are bilingual in Castillian and Galician, so the speed trap signs solemnly switch between "Control de velocidad por radar" and "Control de velocidade por radar". Just in case someone doesn't understand, as several of my friends and in-laws have noted.

 

[pines-demon]

Celerity also exists in English, as far as I know only for the (deservedly obscure) quantity $dx/d\tau$, which is the Lorentz gamma factor times the velocity.

You can use célérité for that too in French.

But I don't think it has the same clear distinction as in English. For example, radar speed trap areas say "Control de velocidad por radar", not rapidez. And in physics I believe you always use velocidad, and would say velocidad vectorial or velocidad magnitud if you need to make the distinction (or so I was told, but by a chemist). I think rapidez is for less quantitative situations. Perhaps @mcastillo356 could comment.

I agree that it is less strict that in English, as one still says "velocidad de luz" (speed of light) instead of "rapidez de la luz", but I guess it can be less or more strict in physics depending on the Spanish speaking region of the world.

 

[Orodruin]

Celerity also exists in English, as far as I know only for the (deservedly obscure) quantity dx/dτ, which is the Lorentz gamma factor times the velocity. I don't think I've seen it used except for people dutifully mentioning its existence before going on to never use it again.

… and then we have rapidity, defined as the arcosh of the inner product of the 4-velocities between observer and object. Unlike velocity, it is still additive in relativity (assuming 1D motion). But this is getting to be an enormous aside to the OP’s question.