## Calculating average velocity or speed Onedimensional motion Physics Khan Academy

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 tutorial, and then i'll add another tutorial 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 51 let me just write it out, 51 kilometers, and you can treat the units the same way you would treat the quantities in a fraction.51 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 tutorial, we talked about things in terms of meters per second.Here, i give you kilometers, or kilometers, 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 13,600 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 13,600 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 tutorial'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.'.

### Finding average speed or rate Ratios proportions units and rates PreAlgebra Khan Academy

Salman khan i have some footage here of one of the most exciting moments in sports history.And to make it even more exciting, the commentator is speaking in german.And i'm assuming that this is ok under fair use, because i'm really using it for a math problem.But i want you to watch this tutorial, and then i'll ask you a question about it.Cheering commentator speaking german salman khan so you see, it's exciting in any language that you might watch it.But my question to you is, how fast was usain bolt going.

What was his average speed when he ran that 100 meters right there and i encourage you to watch the tutorial as many times as you need to do it.And now i'll give you a little bit of time to think about it, and then we will solve it.So we needed to figure out how fast was usain bolt going over the 100 meters.So we're really thinking about, in the case of this problem, average speed or average rate.And you might already be familiar with the notion.

That distance is equal to rate or speed i'll just write rate times time.And i could write times like that, but once we start doing algebra, the traditional multiplication symbol can seem very confusing because it looks just like the variable x.So instead, i will write times like this.So distance is equal to rate times time.And hopefully, this makes some intuitive sense for you.If your rate or your speed were 10 meters per second just as an example.That's not necessarily how fast he went.But if you went 10 meters per second,.

And if you were to do that for two seconds, then it should hopefully make intuitive sense that you went 20 meters.You went 10 meters per second for two seconds.And it also works out mathematically.10 times 2 is equal to 20.And then you have seconds in the denominator and seconds up here in the numerator.I just wrote seconds here with an s.I wrote it out there.But they also cancel out, and you're just left with the units of meters.So you're just left with 20 meters.

So hopefully this makes intuitive sense.With that out of the way, let's actually think about the problem at hand.What information do we actually have so do we have the distance so what is the distance in the tutorial we just did and i'll give you a second or two to think about it.Well, this race was the 100 meters.So the distance was 100 meters.Now, what else do we know do we know well, we're trying to figure out the rate.That's what we're going to figure out.

What else do we know out of this equation right over here well, do we know the time do we know the time what was the time that it took usain bolt to run the 100 meters and i'll give you another few seconds to think about that.Well luckily, they were timing the whole thing.And they also showed that it's a world record.But this right over here is in seconds.It's how long it took usain bolt to run the 100 meters.It was 9.58 seconds.And i'll just write s for seconds.

So given this information here, what you need to attempt to do is now give us our rate in terms of meters per second.I want you to think if you could figure out the rate in terms of meters per second.We know the distance, and we know the time.Well, let's substitute these values into this equation right over here.We know the distance is 100 meters.And it's equal to we don't know the rate, so i'll just write rate right over here.And let me write it in that same color.

It's equal to rate times and what's the time we do know the time.It's 9.58 seconds.And we care about rate.We care about solving for rate.So how can we do that well, if you look at this right hand side of the equation, i have 9.58 seconds times rate.If i were able to divide this right hand side by 9.58 seconds, i'll just have rate on the right hand side.And that's what i want to solve for.So you say, well, why don't i just divide the right hand side.

By 9.58 seconds because if i did that, the units cancel out, if we're doing dimensional analysis.Don't worry too much if that word doesn't make sense to you.But the units cancel out, and the 9.58 cancels out.But i can't just divide one side of an equation by a number.When we started off, this is equal to this up here.If i divide the right side by 9.58, in order for the equality to still be true, i need to divide the left side by the same thing.

So i can't just divide the right side.I have to divide the left side in order for the equality to still be true.If i said one thing is equal to another thing, and i divide the other thing by something, in order for them to still be equal, i have to divide the first thing by that same amount.So i divide by 9.58 seconds.So on our right hand side and this was the whole point these two cancel out.And then on the left hand side, i'm.

Left with 100 divided by 9.58.And my units are meters per second, which are the exact units that i want for rate, or for speed.And so let's get the calculator out to divide 100 by 9.58.So i've got 100 meters divided by 9.58 seconds gives me 10 point this says we've got about three significant digits here so let's say 10.4.So this gives us 10.4.And i'll write it in the rate color.10.4 and the units are meters per second meters per second is equal to my rate.

Now, the next question.So we got this in meters per second.But unfortunately, meters per second, they're not the when we drive a car, we don't see the speedometer in meters per second.We see either kilometers per hour or miles per hour.So the next task i have for you is to express this speed, or this rate and this is his average speed, or his average rate, over the 100 meters.But to think about this in terms of kilometers per hour.So try to figure out if you can rewrite.

This in kilometers per hour.Well, let's just take this step by step.So i'm going to write so let me just go down here, start over.So i started off with 10.4.And i'll write meters in blue, and seconds in magenta.Now, we want to get to kilometers per hour.Right now we're meters per second.So let's take baby steps.Let's first think about it in terms of kilometers per second.And i'll give you a second to think about what we would do this to turn this.

Into kilometers per second.Well, the intuition here, if i'm going 10.4 meters per second, how many kilometers is 10.4 meters well, kilometers is a much larger unit of measurement.It's 1,000 times larger.So 10.4 meters will be a much smaller number of kilometers.And in particular, i'm going to divide by 1,000.Another way to think about it, if you want to focus on the units, we want to get rid of this in meters, and we want a kilometers.So we want a kilometers, and we want to get rid of these meters.

So if we had meters in the numerator, we could divide by meters here.They would cancel out.But the intuitive way to think about it is we're going from a smaller unit, meters, to a larger unit, kilometers.So 10.4 meters are going to be a much smaller number of kilometers.But if we look at it this way, how many meters are in 1 kilometer 1 kilometer is equal to 1,000 meters.This right over here, 1 kilometer over 1,000 meters, this is 1 over 1.We're not changing the fundamental value.

We're essentially just multiplying it by one.But when we do this, what do we get well, the meters cancel out.We're left with kilometers and seconds.And the numbers, you get 10.4 divided by 1,000.10.4 divided by 1,000 is going to give you so if you divide by 10, you're going to get 1.04.You divide by 100, you get 0.104.You divide by 1,000, you get 0.0104.So that's just 10.4 divided by 1,000.And then our units are kilometers per second.So that's the kilometers, and then i.

Have my seconds right over here.So let me write the equal sign.Now, let's try to convert this to kilometers per hour.And i'll give you a little bit of time to think about that one.Well, hours, there's 3,600 seconds in an hour.So however many kilometers i do in a second, i'm going to do 3,600 times that in an hour.And the units will also work out.If i do this many in a second, so it's going to be times 3,600, there are 3,600 seconds in an hour.

And another way to think about it is we want hours in the denominator.We had seconds.So if we multiply by seconds per hour, there are 3,600 seconds per hour, the seconds are going to cancel out, and we're going to be left with hours in the denominator.So seconds cancel out, and we're left with kilometers per hour.But now we have to multiply this number times 3,600.I'll get the calculator out for that.So we have 0.0104 times 3,600 gives us, i'll just say 37.4.So this is equal to 37.4 kilometers per hour.

So that's his average speed in kilometers per hour.And now the last thing i want to do, for those of us in america, we'll convert into imperial units, or sometimes called english units, which are ironically not necessarily used in the uk.They tend to be used in america.So let's convert this into miles per hour.And the one thing i will tell you, just in case you don't know, is that 1.61 kilometers is equal to 1 mile.So i'll give you a little bit of time to convert this into miles per hour.

Well, as you see from this, a mile is a slightly larger or reasonably larger unit than a kilometer.So if you're going 37.4 kilometers in a certain amount of time, you're going to go slightly smaller amount of miles in a certain amount of time.Or in particular, you're going to divide by 1.61.So let me rewrite it.If i have 37.4 kilometers per hour, we're going to a larger unit.We're going to miles.So we're going to divide by something larger than one.So we have one let me write it in blue 1.

Mile is equal to 1.61 kilometers.Or you could say there's one 1.61th mile per kilometer.It also, once again, works out with units.We want to get rid of the kilometers in the numerator.So we would want it in the denominator.We want a mile in the numerator.So that's why we have a mile in the numerator here.So let's once again multiply, or i guess in this case we're dividing by 1.61.And we get let's just divide our previous value by 1.61.And we get 23 point i'll just round up 23.3.

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