Is velocity ever a scalar quantity?

[paulb203]

Temporary divergence [Edit - I mean digression] and reminisce…

When teaching/introducing this stuff (a very long time ago) I would would walk one complete circuit around the lab’ and ask the students to write down (anonymously, on a scrap of paper) their own estimates for the distance covered, time taken, displacement, average speed and average velocity.

We’d collect the scraps and skim through the (disconcertingly wide range of) answers. Then we’d go through the ‘correct’ answers.

It was a very useful teaching exercise as you could easily assess the extent of (mis)understanding and (though less important) mis-estimation.

The nice thing was that individual students were actually engaged – wanting to compare their own results to the rest of the class’s.

Keeping it anonymous was a nice touch.

 

[jtbell]

Regards that last bit, where you say average velocity IS NOT EQUAL TO the magnitude of average velocity,

That last equation says average speed is not (generally) equal to the magnitude of average velocity.

 

[Steve4Physics]

Keeping it anonymous was a nice touch.

I'm baffled - why do you say 'anonymous'?

(Unfortunately there is no available emoji for 'baffled'!)

Ignore! I'm being dumb.

 

[Orodruin]

That last equation says average speed is not (generally) equal to the magnitude of average velocity.

… and, more generally, the integral of an absolute value is not equal to the absolute value of the integral.

 

[robphy]

Yes, velocity by definition is a derivative. The only ambiguity comes when it is constant. And then velocity and average velocity are the same.

An alternative notation for average velocity is: $\vec v_{avg} \equiv \frac{\Delta \vec x}{\Delta t}$. Average speed is harder to define formally. It would have to be:
$$v_{avg} \equiv |\vec v|_{avg} = \frac 1 {\Delta t}\int_{t_0}^{t_0 + \Delta t} |\vec v(t)| dt$$And something very important is that:
$$v_{avg} \neq |\vec v_{avg}|$$

For completeness, and in order of importance,

• [instantaneous] velocity $$\vec v(t) \equiv \frac{d}{dt}\vec x(t)$$
• [instantaneous] speed is the magnitude of the velocity-vector (single bar is also okay)
  $$\left\|\vec v(t)\right\|\equiv\sqrt{\vec v(t) \cdot \vec v(t)}= \left\|\frac{d\vec x}{dt} \right\| $$[update] which is sometimes written as "v" (which is \vec v without its \vec{\quad } arrowhead).
  
  
• [time-weighted] average-velocity [over an interval]
  the "time-weighted-average" of the velocity-vector $$\vec v_{avg}(t_0,t_1) \equiv \frac{\int_{t_0}^{t_1} \vec v(t) dt}{\int_{t_0}^{t_1} dt} = \frac{\Delta \vec x}{\Delta t}=\frac{\mbox{total displacement-vector}}{\mbox{total elapsed time}}$$[update]average-velocity as total-displacement/total-elapsed-time is secondary---but it's why this construction is physically useful
  
  
  
• [time-weighted] average-speed [over an interval]
  the "time-weighted-average" of the speed $$v_{avg}(t_0,t_1)=\left\|\vphantom{\|}\vec v\right\|_{avg}\equiv \frac{\int_{t_0}^{t_1} \left\|\vec v(t)\right\| dt}{\int_{t_0}^{t_1} dt} = \frac{ d_{total} }{\Delta t}=\frac{\mbox{total distance}}{\mbox{total elapsed time}}$$
  [update]average-speed as total-distance/total-elapsed-time is secondary---but it's why this construction is physically useful
  

I try to define velocity first.
(I really dislike defining average-velocity before defining velocity, as done in many textbooks.)

---

update:

• the x-component of the [instantaneous] velocity $$v_x(t)=\hat x\cdot \vec v(t),$$
  which is sometimes just written as "v" in introductory textbook chapters dealing with 1-dimensional motion. This notation ("v") contributes to the confusions students have distinguishing the velocity-vector, the x-component of that vector, and the speed (the magnitude of that vector).

 

[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”)

 

[robphy]

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

 

[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.

 

[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.

 

[berkeman]

Thread closed for Moderation.

 

[PeterDonis]

A subthread based on misinformation has been deleted. Thread reopened.

 

[robphy]

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.

As I said earlier ( #35 where I gave a sequence of definitions)

I try to define velocity first.
(I really dislike defining average-velocity before defining velocity, as done in many textbooks.)

To continue my rant, I often follow up with
"How can you define the average of something before defining the something first?"

It's clear that the physics-textbooks are following the math-textbooks process
of defining the tangent-line as the limit of secant-lines....
but math folks don't likely encounter a similar problem since their terms are distinct
(the slope of the secant line isn't described as an "average slope-of-the-tangents").

Furthermore, textbooks don't seem to explain that "average-velocity" is a time-weighted-average of velocities, which is not generally a straight-average.

In short, "average-velocity" should be demoted...

• never introduced first
• never presented to appear more important than [instantaneous] velocity
• should be defined in terms of velocity (to emphasize that "average-velocity" is not the same thing as "velocity", and is less fundamental)

my $0.02.

 

[john1954]

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.

v bold or v with an arrow are vector velocity. If v is not bold or does not have an arrow it is the magnitude of the velocity, called the speed.

 

[Orodruin]

v bold or v with an arrow are vector velocity. If v is not bold or does not have an arrow it is the magnitude of the velocity, called the speed.

While this is usual convention, you cannot always trust that it is being used by all authors. It is therefore best not to always implicitly assume this.

 

[PeroK]

v bold or v with an arrow are vector velocity. If v is not bold or does not have an arrow it is the magnitude of the velocity, called the speed.

This convention is generally not followed in the introductory SUVAT equations, where $v$ is the velocity vector.