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Vertical Motion Example 1
Hi, everyone!
Welcome back to integralcalc.com.
Today we're going to be doing a vertical motion problem.
And in this one, we've been given this word problem.
A ball is thrown straight upward from the ground with an initial velocity of v sub zero equals ninety-six feet per second.
Assuming constant gravity, find the maximum height that the ball attains, as well as its velocity when it hits the ground.
So inorder to solve this problem, we're going to need two things, initially.
One is a diagram that is going to organize all of the information we've been given in the word problem,
and the second thing is the position function.
We haven't been given a function in our word problem so we're going to need to write one that's going to model the position function.
So again the first thing we need is a diagram so let's make our diagram on the x,y coordinate axis
And this is really bad but we've got x and y.
So we can say that our ball takes a flight path that's going to look roughly like this.
So it's moving this way, right?
And we can write the following information about this vertical motion problem.
So at each point, and let's go ahead and mark these...
at the point where the ball is first thrown,
at the point where the ball attains its maximum height
and at the point where it hits the ground.
There are three things that we can find about this function.
One is the time at which the event occurs, the other is the height of the ball and the third thing is the velocity of the ball.
so we know...
and let's go ahead and write that for all three.
So we know a couple things from our word problem.
One, we know that the initial velocity is ninety-six feet per second.
So we can say that the initial velocity is ninety six.
We also know that time is zero because the position function doesn't start until the ball is thrown.
So right before the ball is thrown, time t is zero.
We also know a couple things.
That the hieght of the ball right before it's thrown is zero as long as they say that the ball is thrown straight upward from the ground, right?
Since it's from the ground, the height of the ball is zero.
If it were from a height of 50 feet then y would be fifty here.
But the height is zero.
The height is also going to be zero when the ball hits the ground.
And the only other thing we know is that the velocity when the ball reaches its maximum height is going to be zero.
Because rememebr that velocity indicates two things.
It indicates the speed of the object but speed with direction.
So since the ball is moving up on this side, right?
And then moving down on this side, it's maximum height,
the very top point at which it reaches its maximum height here, it's neither moving up nor down.
It’s finished moving up but has not yet started moving down which means that its velocity is going to be zero.
So we filled in everything we can here.
This is going to be our diagram.
And we have been asked, remember, to find the maximum height the ball attains,
so we're going to be looking for the maximum height, y.
and at once, velocity when it hits the ground.
So it also wants this here, velocity when it hits the ground.
So those are the two things we need to find.
So now we've got our information organized, we need a function that's going to model the position of our ball.
And the position function will be y of t.
and we can use this formula over here to write down our position function.
So, a couple things about this formula.
The first is that if our problem tells us to assume constant gravity as it does,
that means that g in our formula here is going to be thirty-two feet per second.
Our problem is in feet per second so we'll use thirty-two.
If you've been given meter's then it's going to be nine point eight meters per second
but we'll be plugging in thirty two for g because we're working in feet per second.
So thirty-two for g and then the other interesting thing about his function is that it gives us the initial velocity and the initial height here.
So given that that's our formula and we've got a couple of those things,
we can easily write the position function as negative one-half.
We'll plug in thirty-two for g, we're leaving the variable t.
So we leave t squared and then we add to that...
This is the initial velocity, which the problem told us was ninety-six
so we have ninety-six t and the initial height was sub zero which the problem told us was zero
because the ball is being thrown upward from the ground.
so the initial height is zero.
So if we simplify this, negative one-half times thirty-two is going to give us negative sixteen t squared plus ninety-six t.
so this is now y of t, our position function.
So now that we have our position function and our diagram, we can go ahead and try to find the maximum height of the ball
and the velocity when it hits the ground; the two things we've been asked for.
So the only thing we know about the maximum height of the ball is that at that time our velocity is zero.
So we need to find a way to use that piece of information that velocity is zero inorder to find the maximum height.
Well, we need to remember that the velocity function is the derivative of the position function.
So if we want to find...
if we want to use this fact that velocity is equal to zero here,
we can take the derivative of the position function to get the velocity function, vt and set that equal to zero.
So we will find the derivative of the position function which is going to be y prime of t.
That's the same thing as the velocity function.
So the derivative of the position function is the velocity function which is negative thirty-two t plus ninety-six, right?
We just took the derivative of this function here.
So negative thirty-two t plus ninety-six.
If we set that equal to zero, equals zero because we know that the velocity here is equal to zero,
then we can add thirty-two t to both sides and get ninety-six equals thirty-two t.
We'll divide both sides by thirty-two to solve for t and we'll get t equals three.
So now we know that the ball reaches its maximum height at time t equals three.
So we can go ahead and add that information to our diagram but we're really looking for the height.
The maximum height.
And in order to find that, we'll plug t equals three into our original position function which is this function here that we simplified, right?
So we plug t equals three into our position function y so we get y of three equals negative sixteen times three squared,
three squared plus ninety-six times three and when we simplify that we'll get negative sixteen times...
negative sixteen times nine, which is going to be negative one forty-four.
And then ninety-six times three which is going to be two eighty-eight.
And of course, negative one forty-four plus positive two eighty-eight gives us a positive one forty-four.
So one forty-four is the maximum height that the ball attains.
So we've solved for one-half of our problem. So y equals one forty-four.
Now, we need to find the velocity when the ball hits the ground.
So again, in order to do that, we can set our position function equal to zero, right?
Because we only know that y is equal to zero.
So we'll use our original position function y, set that equal to zero and solve for velocity that way.
So we'll plug in... We'll set the position function equal to zero.
This tells us again that... zero equals t squared...
that the height of the ball at this time is zero, which of course when it hits the ground, height would be zero.
So we set that equal to zero and then we can factor out a t.
so we'll get t times negative sixteen t plus ninety six and when we solve this for t,
we know that either t is going to equal zero or if we solve this part here for t,
we'll say zero equals negative sixteen t plus ninety-six,
we'll add sixteen t to both sides and get sixteen t equals ninety six.
Dividing both sides by sixteen, it gives us t equals six.
So t is going to either equal zero or six which makes sense because the height of the ball at time t equals zero is zero.
so we got that's this solution over here
and we know that the other time that the height is zero is going to be when the ball hits the ground which is t equals six.
So we've now found that part.
Now we just need to find velocity.
So since we know that t is equal to six, we can use our velocity function here and plug in six for t to solve for velocity.
So we'll say velocity at time t equals six when it hits the ground is going to be negative thirty-two times six,
plugging in six for t again, plus ninety six. And when we simplify that, negative thirty two times six gives us...
Let's see... one eighty and twelve would be one-ninety two,
negative one ninety-two plus ninety six is going to be a positive... negative..
er sorry.. a negative ninety six, so what this tells us...
velocity is negative ninety-six.
The reason that velocity is negative because velocity indicates direction.
So our first velocity here positive ninety-six, means that the ball was thrown upward with the speed of 96 feet per second.
The negative ninety-six here indicates that the ball is coming straight down at a speed of ninety six feet per second.
So it's positive and negative because its indicating direction.
So our final answers are going to be that the maximum height of the ball is one hundred and forty-four feet
and then the velocity when the ball hits the ground is negative ninety six feet per second.
So that's it. I hope that video helped you, guys. And I will see you in the next one.
Bye!