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This is the average speed of the ball. Now, of course, I took the ball and I

threw it down with some force and then it came back up and hovered here and then

came back down if you recall. So, what I'm trying to say is the speed

of the ball was changing the entire time and this is the average speed and so

sometimes the speed was faster and sometimes it was slower.

Here. So, think about when I just threw the

ball down. From leaving my hand to touching the

ground, that was about 2. seconds. And the distance it traveled was about

one meter. One meter divided by 2. seconds is 5

meters per second. So we found the average speed to be 2.85

meters per second. But that's not really the instantaneous

speed of the ball. Because when I threw down the ball, the

ball was traveling much faster. And then the ball bounced up.

And then it started slowing down. And then it slowed down to almost no

speed at all. And then it came back into my hand going

a little faster. So how do we figure out how fast the

balls going at say one of those other time intervals.

We found the average speed over the course of the entire trajectory the full

1.1 seconds to be 2.85 meters per second. However if we only consider the first 2.

seconds we find the average speed to be five meters per second, this makes sense

because if you watch the video you see the ball is moving rather quickly in the

first 2. seconds of its trajectory. You probably asking yourself, how does

Bart know that the ball traveled one meter during the first 2. seconds of its

journey and 3.14 meters during the whole of 1.1 second journey down up and down?

Well, we know this because we got the video.

Alright, here Bart is just about to release the ball from his hand.

And 1, 2, 3, 4, 5, 6 frames later, the ball hits the ground.

There's 30 frames being shot in every second so that means that it took 6 / 30

or 2. seconds for the ball to hit the ground.

Making a bunch more measurements, I can combine all this information in a graph.

This is a graph of a function the input to this function is time so along the x

axis I'm plotting time and seconds. The output to the function is the height

the height of the ball at that particular moment in time.

Now, on this graph, I can go back now and try to figure out how fast the ball is

moving at say 4. seconds after Bart releases it and at 8. seconds after Bart

releases it. So let's try to figure out how fast the

ball is moving at 4. seconds. Here I've marked the position of the

ball, .4 seconds after Bart releases the ball.

Now we don't really have any way of figuring how fast the ball is moving at

that particular moment. What I can do is figure out the average

speed of the ball during some time interval.

So as sort of a first guess to how fast this thing is moving at 4. seconds, I'm

going to figure out the average speed of the ball between 4. seconds and 6.

seconds. Alright, so I've got this handy table

here, of function values at 4. seconds. The the ball was 101.1 centimeters above

the ground and 6. seconds after Bart released the ball, the ball was 16.8

centimeters above the ground. So I can put that information together to

figure out the speed of the ball between 4. and 6. seconds.

Alright. So

5:40

4. seconds to say 6. seconds at 4. seconds on my chart the ball was a 101.1

centimeters above the ground. At 6. seconds the ball was a 161.8

centimeters above the ground. Now this time interval has a length of 2.

seconds. And how far did the ball move during that

time interval? Well 161.8 - 101.1 is 60.7 centimeters.

So during the 2. seconds that elapsed from 4. seconds to 6. seconds after Bart

released the ball, the ball traveled a distance of 60.7 centimeters, which means

the speed, which is the distance traveled over time is 60.7 over 2. centimeters per

second, which is 303.5 centimeters per second or 3.035 meters per second.

Which is about seven miles per hour. Now I could do a little bit better,

alright? Here I'm calculating the average speed of

the ball between 4. and 6. seconds, but I'm trying to figure out how fast the

ball is moving at this particular moment. So instead of just calculating the

average speed during this time interval to be about seven miles per hour, I could

do it over a shorter time interval, all right?

Instead of 6. to 4,. I could go from 4. to say 5..

Here's half a second after Bart released the ball.

And I could figure out the average speed of the ball during this part of its

trajectory. Let's see how we calculate that.

Well, it's the same kind of game. All right?.4 seconds after Bart released

the ball, the ball was 101.1 centimeters above the ground.

5. seconds after Barb released the ball, looking back at my table I find that the

ball was 136.5 centimeters above the ground.

136.5 centimeters. This time interval was 1. seconds long.

And how far did the ball move during that time interval?

Well, that's 35.4 centimeters. 136.5 minus 101.1 is 35.4.

So, the ball moved 35.4 centimeters. During the point one seconds that elapsed

point four seconds to 5. seconds after Bart released the ball.

Speed is how far you've traveled over how long it took you so if I divide these

this is the speed of the ball the average speed of the ball between point four and

point five seconds and this works out to be 354 centimeters per second I mean I'm

dividing by this very nice number point one.

which is the same as 3.45 meters per second which is about eight miles per

hour. And indeed, if you look back at this,

this chart, between 4. and 6. seconds, yeah.

Maybe the average speed was u, about seven miles per hour.

Between 4. and 5. seconds the average speed was a little bit higher.

You know, the average speed here worked out to be eight miles per hour instead of

seven miles per hour. The average speed of the ball during this

time interval is higher than during this whole time interval.

We're still not there. We are trying to figure out how fast the

ball is moving at this particular moment right not the average speed between 4.

and 5. seconds. To get closer, right, we should take an

even smaller time interval. Instead of 4. to 5. well, why not look

back on our handy chart here and see well, here is where the ball is at 4.

seconds. Here's where the ball is at 42. seconds.

We could use this information to figure out the speed of the ball just during the

very tiny time interval between 4. and 42. seconds after Bart releases the ball.

Well, let's do that. All right.

So again, .4 seconds after Bart releases the ball, the ball is 101.1 centimeters

above the ground. .42 seconds after Bart releases the ball,

the ball is 109 centimeters above the ground, that's what this chart is telling

me .42 seconds after Bart releases the ball, 109 centimeters above the ground.

So that means during the very tiny time interval 02. seconds that elapsed between

4. seconds and 42. seconds after Bart releases the ball, the ball has traveled

how far well 109 - 101.1 centimeters is just 7.9 centimeters.

So in two hundredths of a second the ball has traveled 7.9 centimeters.

To figure out the speed I again divide 7.9 / 02. is 395 centimeters per second.

Which is 3.95 meters per second. Which is about 9 miles per hour.

This is a much better approximation to the instantaneous speed of the ball at 4.

seconds. Look, here's the graph again.

Between point four and point six seconds, the ball is travelling maybe seven miles

per hour on average. Between point four and point five

seconds, the ball is travelling maybe. Eight miles per hour between 4. and 42.

seconds we just calculated that the speed of the ball is about nine miles per hour.

And that makes a whole lot of sense. Right?

The speed of the ball from here to here is slower than the average speed from

here to here. Which is slower than average speed from

0.4 to 0.42 seconds. The ball's slowing down in its

trajectory, so the average speed over these shorter time intervals is

decreasing. So, let's figure out how fast the ball

was moving at 8. seconds. That's when the ball was at the top of

its trajectory. we can't really do that.

All I can really do is figure out the average speed of the ball over some time

interval but I've got a table of values of the function.

And I know how high the ball was at 8. seconds after release.

It was 182 centimeters, and I can compute its average speed over a very short time

interval, like the time interval between 8. and 81. seconds after release, all

right? And the ball didn't move very far during

that time interval but of course that time interval also isn't very long, so

it's not super clear how fast the ball might be moving on average during that

time interval. We can do the calculation though.

Let's do it now, so. 0.8 seconds after the ball was released,

the ball was 182 centimeters above the ground.

0.81 seconds after the ball was released, the ball was 181.9 centimeters above the

ground. Now this time interval between 8. and 81.

seconds has the duration of just one hundredth of a second.

81. - 8. is 01.. That's a very short amount of time and

during that short amount of time, how far do the ball move?

Well, 181.9 - 182 centimeters, that's just.1 centimeters.

And if we're being pedantic, it's negative 1. centimeters.

All right? The ball fell between 8. and 81. seconds,

so this number is recording not only how far it moved but also the direction that

it moved in. It's really displacement instead of a

distance. Anyhow, .01 centimeters divided by 01.

seconds, that will give me the velocity, right?

Displacement over time. So if I divide these, this ratio here is

10 centimeters per second or -10 if I am keeping track of the direction its moving

in. It's falling down at a speed on average

of ten centimeters per second during this time interval.

That's -.1 meters per second. Which is about.2 miles per hour or -.2

miles hour if I'm keeping track of the direction it's going.

Anyway, .2 miles per hour is a really slow speed,

right? The ball is not moving very much on

average between 8. and 81. seconds. In light of this, it might make sense to

say that the instantaneous speed of the ball at.4 seconds is nine miles per hour.

Now, why? Well, the average speed between 4. and 6.

seconds is maybe 7 miles per hour. The average speed from 4. and 5. is about

eight miles per hour. The average speed between 4. and 42.

seconds is about 9 miles per hour. You know and based on this,

it seems like if we took a really short time interval, just after 4. seconds and

tried to calculate how fast the ball was going on average during that very small

time interval, you might conclude that the average speed during a very small

time interval is about 9 miles per hour. It's in that sense that we're going to

say that the instantaneous speed of the ball at point four seconds is 9 miles per

hour. When you play the same game 8. seconds

into the balls journey, alright. When it's just to the top of its

trajectory. So the average speed of the ball, between

8. and 81. seconds is exceedingly slow, and you can see that in the video,

alright. The ball is barely moving, at the top of

its trajectory. What's the instantaneous speed of the

ball at the top of its trajectory? It's zero, right?

I mean yes. The average speed over a time interval

between 8. and 8000001. seconds isn't zero.

But if you look at an average speed over an exceedingly small time interval, those

average speeds over shorter and shorter time intervals are as close to zero as

you like. That's the sense in which the

instantaneous velocity at the top of the trajectory, the limit of the average

velocities over small time intervals, is zero.

Isn't calculus amazing? We're using the idea of limits to compute

instantaneous speed. Using a little bit of math, we can

understand the world around us. That's the power of calculus.

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