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### Course: AP®︎/College Physics 1 > Unit 1

Lesson 1: Position, acceleration, and velocity# Calculating average velocity or speed

Although speed and velocity are often words used interchangeably, in physics, they are distinct concepts. Velocity (v) is a vector quantity that measures displacement (or change in position, Δs) over the change in time (Δt), represented by the equation v = Δs/Δt. Speed (or rate, r) is a scalar quantity that measures the distance traveled (d) over the change in time (Δt), represented by the equation r = d/Δt. Created by Sal Khan.

## Want to join the conversation?

- what is the difference between speed and velocity?(224 votes)
- There is not that much of a difference, but there are a few very important but subtle ones.

1) Speed is a scalar (only magnitude) and Velocity is a vector (magnitude AND DIRECTION)

2) You will only have a positive speed, but velocity can be negative if you are moving in the negative direction (which depends on the coordinate axes you use)

3) Speed basically tells how fast something is moving, but velocity tells us the rate which an object changes position (delta x divided by delta t, or dx/dt if you know derivatives)

Check out this site: http://www.physicsclassroom.com/class/1dkin/u1l1d.cfm

It should help with your question if I don't.

Hope it helps :)(556 votes)

- could it be that we use S for displacement because of the Latin word spatium which means distance?(94 votes)
- Seems highly probable.(26 votes)

- What is the actual purpose of having a velocity ? Like in real life problems we deal with speed, I am moving with a speed of this and If I reach to the same point where I started from and If someone says you have an avg velocity is zero, It sound so weird, I mean I spent a lot of energy and time from travelling from one point to other and I find my vel is zero and speed is something. In short What was the purpose to bring the velocity concept in physics?(40 votes)
- Thats a nice question....

In a way, you are asking the question "what is the point in vectors...?"

vectors contain more information than scalars and the relative directions velocity become very important when dealing with the next level (or derivative) acceleration. Acceleration is the change in velocity over the time taken to make the change. This will, then, be influenced by the angle between the final and initial velocities.

Kinetic theory:

when we consider the average velocity of particles in a gas, we find that it, too , is zero. whereas the mean kinetic energy of all the particles in the gas is non-zero because it is related to the average velocity (squared).

hope that helps(52 votes)

- If the problem indicated that Shantanu traveled 5 km north and then 4 km south, would the average velocity be 1 km/hour or 9 km/hour. i.e would you use the distance traveled or displacement?(18 votes)
- You would have to use the distance traveled. For example, if you were stopped by a police officer, going 70 mph (average) in a 55 mph zone, he/she would not let you get away if you just reversed quickly, and then said that your average speed now is 55 mph.(15 votes)

- is there a difference between magnitude and measurement?(15 votes)
- Magnitude is the relative size or extent but measurement is the numerical amount.(14 votes)

- The letter "s" is commonly used to denote displacement because of its historical origins. In physics, the symbol "s" for displacement comes from the Latin word "spatium," which means space(13 votes)
- at4:57, why does Sal say the average speed is 5 kilometers per hour to the north? Shouldn't it just be 5 kilometers per hour because it's the same speed if you are going south or east or west?(7 votes)
- If you were referring to speed, you would be right, but since we are dealing with velocity, a *vector,* which in a previous video he explained that a vector has a position/size and a *direction.* Speed is a schalar, which he is not using, which does not have a direction.(11 votes)

- At3:46, what is constant velocity?(6 votes)
- Constant velocity here means that his velocity was not changing so neither his speed nor his direction would change, everything would remain constant.(7 votes)

- i can't understand what is the point of velocity? like for example if i go jogging i would want to know what my average speed is..why would i want to know the velocity?

pls answer(5 votes)- Hi Viveka,

You don't know this yet but you have just asked a very important question. A question that will change the way you view the world and how you look at mathematics.

Suppose you are the driver of a race car. Now suppose you modify this car with a rocket assist. I think you will agree that the rocket assisted car will have a greater acceleration.

But wait, what is acceleration? Acceleration is defined as the rate of change in velocity. In the rocket assisted car the velocity is changing very fast. You will be "pushed" forcefully back into the seat as you drive this car. Stated another way you will go from 0 to 60 very quickly.

I could go on but I think you see the point. Without the concept of velocity you can't talk about more interesting things like acceleration.

Know that there is an entire field of mathematics devoted to the study of things in motion.

I wish you success in calculus.

Regards,

APD

P.S. Please leave a comment below if you have any questions.(5 votes)

- s = spostamento , in lingua italiana. maybe..(4 votes)
- According to convention, we use "s" for displacement, after the latin word "spacium"—same idea though!(7 votes)

## Video transcript

Now that we know a little bit
about vectors and scalars, let's try to apply what we
know about them for some pretty common problems you'd, one,
see in a physics class, but they're also common problems
you'd see in everyday life, because you're trying to
figure out how far you've gone, or how fast you're
going, or how long it might take you to
get some place. So first I have, if
Shantanu was able to travel 5 kilometers north in
1 hour in his car, what was his average velocity? So one, let's just
review a little bit about what we know about
vectors and scalars. So they're giving us that
he was able to travel 5 kilometers to the north. So they gave us a magnitude,
that's the 5 kilometers. That's the size of
how far he moved. And they also give a direction. So he moved a distance
of 5 kilometers. Distance is the scalar. But if you give the direction
too, you get the displacement. So this right here
is a vector quantity. He was displaced 5
kilometers to the north. And he did it in
1 hour in his car. What was his average velocity? So velocity, and
there's many ways that you might see it defined,
but velocity, once again, is a vector quantity. And the way that we
differentiate between vector and scalar quantities
is we put little arrows on top of vector quantities. Normally they are bolded,
if you can have a typeface, and they have an
arrow on top of them. But this tells you
that not only do I care about the
value of this thing, or I care about the
size of this thing, I also care about its direction. That's what the arrow. The arrow isn't
necessarily its direction, it just tells you that
it is a vector quantity. So the velocity of something
is its change in position, including the direction
of its change in position. So you could say
its displacement, and the letter for
displacement is S. And that is a
vector quantity, so that is displacement. And you might be
wondering, why don't they use D for displacement? That seems like a much
more natural first letter. And my best sense of that is,
once you start doing calculus, you start using D for
something very different. You use it for the
derivative operator, and that's so that the
D's don't get confused. And that's why we use
S for displacement. If someone has a better
explanation of that, feel free to comment
on this video, and then I'll add another
video explaining that better explanation. So velocity is your
displacement over time. If I wanted to write an
analogous thing for the scalar quantities, I could
write that speed, and I'll write
out the word so we don't get confused
with displacement. Or maybe I'll write "rate." Rate is another way that
sometimes people write speed. So this is the vector version,
if you care about direction. If you don't care
about direction, you would have your rate. So this is rate, or speed,
is equal to the distance that you travel over some time. So these two, you could
call them formulas, or you could call them
definitions, although I would think that they're
pretty intuitive for you. How fast something is
going, you say, how far did it go over some
period of time. These are essentially
saying the same thing. This is when you
care about direction, so you're dealing with
vector quantities. This is where you're not so
conscientious about direction. And so you use distance,
which is scalar, and you use rate or
speed, which is scalar. Here you use displacement,
and you use velocity. Now with that out of the
way, let's figure out what his average velocity was. And this key word,
average, is interesting. Because it's possible that
his velocity was changing over that whole time period. But for the sake of
simplicity, we're going to assume that it was
kind of a constant velocity. What we are calculating is going
to be his average velocity. But don't worry about
it, you can just assume that it wasn't changing
over that time period. So his velocity is,
his displacement was 5 kilometers to the north--
I'll write just a big capital. Well, let me just write it
out, 5 kilometers north-- over the amount of
time it took him. And let me make it clear. This is change in time. This is also a change in time. Sometimes you'll just
see a t written there. Sometimes you'll see
someone actually put this little triangle,
the character delta, in front of it, which
explicitly means "change in." It looks like a very fancy
mathematics when you see that, but a triangle in
front of something literally means "change in." So this is change in time. So he goes 5 kilometers
north, and it took him 1 hour. So the change in
time was 1 hour. So let me write that over here. So over 1 hour. So this is equal
to, if you just look at the numerical
part of it, it is 5/1-- let me just write
it out, 5/1-- kilometers, and you can treat the
units the same way you would treat the
quantities in a fraction. 5/1 kilometers per hour,
and then to the north. Or you could say this
is the same thing as 5 kilometers per hour north. So this is 5 kilometers
per hour to the north. So that's his average velocity,
5 kilometers per hour. And you have to be careful,
you have to say "to the north" if you want velocity. If someone just said "5
kilometers per hour," they're giving you a speed,
or rate, or a scalar quantity. You have to give the direction
for it to be a vector quantity. You could do the same
thing if someone just said, what was his average
speed over that time? You could have said, well, his
average speed, or his rate, would be the
distance he travels. The distance, we don't care
about the direction now, is 5 kilometers, and
he does it in 1 hour. His change in time is 1 hour. So this is the same thing
as 5 kilometers per hour. So once again, we're only
giving the magnitude here. This is a scalar quantity. If you want the vector, you
have to do the north as well. Now, you might be saying,
hey, in the previous video, we talked about things in
terms of meters per second. Here, I give you kilometers,
or "kil-om-eters," depending on how you
want to pronounce it, kilometers per hour. What if someone wanted
it in meters per second, or what if I just wanted to
understand how many meters he travels in a second? And there, it just becomes
a unit conversion problem. And I figure it doesn't hurt
to work on that right now. So if we wanted to do
this to meters per second, how would we do it? Well, the first step is to
think about how many meters we are traveling in an hour. So let's take that 5
kilometers per hour, and we want to
convert it to meters. So I put meters
in the numerator, and I put kilometers
in the denominator. And the reason why I do that
is because the kilometers are going to cancel out
with the kilometers. And how many meters are
there per kilometer? Well, there's 1,000 meters
for every 1 kilometer. And I set this up right here so
that the kilometers cancel out. So these two
characters cancel out. And if you multiply,
you get 5,000. So you have 5 times 1,000. So let me write this-- I'll do
it in the same color-- 5 times 1,000. So I just multiplied
the numbers. When you multiply something,
you can switch around the order. Multiplication is
commutative-- I always have trouble pronouncing
that-- and associative. And then in the units, in the
numerator, you have meters, and in the denominator,
you have hours. Meters per hour. And so this is equal to
5,000 meters per hour. And you might say,
hey, Sal, I know that 5 kilometers is the
same thing as 5,000 meters. I could do that in my head. And you probably could. But this canceling out
dimensions, or what's often called
dimensional analysis, can get useful once you
start doing really, really complicated things with less
intuitive units than something like this. But you should always do an
intuitive gut check right here. You know that if you do
5 kilometers in an hour, that's a ton of meters. So you should get
a larger number if you're talking
about meters per hour. And now when we want
to go to seconds, let's do an intuitive gut check. If something is traveling a
certain amount in an hour, it should travel a much
smaller amount in a second, or 1/3,600 of an hour, because
that's how many seconds there are in an hour. So that's your gut check. We should get a smaller
number than this when we want to say
meters per second. But let's actually do it with
the dimensional analysis. So we want to cancel
out the hours, and we want to be left with
seconds in the denominator. So the best way to cancel
this hours in the denominator is by having hours
in the numerator. So you have hours per second. So how many hours
are there per second? Or another way to
think about it, 1 hour, think about the larger unit,
1 hour is how many seconds? Well, you have 60 seconds
per minute times 60 minutes per hour. The minutes cancel out. 60 times 60 is 3,600
seconds per hour. So you could say this is 3,600
seconds for every 1 hour, or if you flip them, you would
get 1/3,600 hour per second, or hours per second, depending
on how you want to do it. So 1 hour is the same
thing as 3,600 seconds. And so now this hour
cancels out with that hour, and then you multiply,
or appropriately divide, the numbers right here. And you get this is equal
to 5,000 over 3,600 meters per-- all you have left in the
denominator here is second. Meters per second. And if we divide both the
numerator and the denominator-- I could do this by hand, but
just because this video's already getting a
little bit long, let me get my trusty
calculator out. I get my trusty calculator
out just for the sake of time. 5,000 divided by 3,600, which
would be really the same thing as 50 divided by
36, that is 1.3-- I'll just round it
over here-- 1.39. So this is equal to
1.39 meters per second. So Shantanu was traveling
quite slow in his car. Well, we knew that just
by looking at this. 5 kilometers per hour, that's
pretty much just letting the car roll pretty slowly.