Here we are at a lecture hall at The Ohio State University.
I'm Bart Snap. Today, we're going to see how we can use
Calculus to study the path of an object through space, in particular this orange
ball.
When the ball leaves my hand, it's going pretty fast.
But as it approaches the ground, it's going faster and faster, and faster until
it hits the ground, bounces up at nearly the same speed.
And then it's going quickly, but slower, slower, slower, until, it reaches its
pinnacle and then it stopped. And then it starts falling back down
again, slowly at first, but then quicker, quicker and quicker, and then I, grab the
ball. So how fast was the ball moving well lets
see. The total distance the ball traveled was
about, 3.14 meters and the total time the ball traveled for, was about 1.1 seconds.
Dividing this, we find this is equal to 2.85 meters per second.
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
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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