00:01
So for this question, we're given the trajectory of a ball thrown vertically upwards with an initial velocity of 80 feet per second.
00:09
It's given by s of t equals 80t minus 16 t squared.
00:15
And we're asked a few questions about the trajectory of this ball.
00:19
First, we want to find the maximum height reached by the ball.
00:23
That is going to be when the derivative of this function, which is equal to the velocity of, of this function is equal to zero.
00:35
And if you want some physical intuition behind this, let's say the trajectory of our ball looks roughly like this.
00:43
It is a parabola that's pointed downwards, since this coefficient here, this minus 16 is negative.
00:51
And you can easily see one of its roots is zero.
00:57
So why is its maximum when the derivative is equal to zero? well the derivative gives the tangent line the slope of the tangent line at a point and if that slope is equal to zero then we see that this thing is at its maximum so that's when the derivative is equal to zero so using this we just need to differentiate this function set it equal to zero and solve for t so let's do that now um differentiating we get uh d s d t is equal to 80 minus 32 t.
01:37
And then we want this to be equal to zero.
01:41
So now we can rearrange and get t equals 80 over 32 seconds, since t is in seconds, which is equal to 2 .5 seconds.
01:59
So that's when the ball reaches its maximum height.
02:04
After 2 .5 seconds.
02:07
Next, we want to know what the velocity of the ball is when it's 96 feet above the ground on its way up and down.
02:16
So first, we need to find the time at which the ball is 96 feet above the ground.
02:26
So if s is equal to 96, that's 80t minus 16 t squared.
02:33
We need to solve this equation for t.
02:35
So rearranging, i get 16 t squared minus 80t plus 96 is equal to zero.
02:45
This is just a quadratic equation.
02:47
We know how to solve those.
02:49
Let's first divide everything by 16 to get t squared minus 5t plus 6 equals zero.
02:58
And we can just factor this.
03:00
We get t minus two times t minus three.
03:09
Equals zero.
03:10
Let's just check that we expand this out.
03:12
We get the same thing...