00:01
Okay, so in this problem, we got the position function, basically, a particle traveling a straight line, and its displacement is modeled in this graph, and it gives us the function, s of t, equals 8t minus t squared.
00:20
Okay, and it tells us it starts at the origin, 0 ,0, and we want to know when does it go through the origin again.
00:29
So that means we know that when we put zero in for t, we know it's at the point zero comma zero.
00:38
Well, we want to know what's the p value where we hit zero again.
00:43
Well, if we set this equal to zero and factor out a t, not eight, just factor out the t, eight minus t, well, this gives us the zero.
00:57
That gives us this point.
00:59
This factor where t equals eight that's how you find your p value so the value of p is eight seconds i guess yeah seconds okay so that's when it passes through the origin again okay now says particle a changes direction when t equals q write down the value of q show that the particle changes direction at q.
01:31
Okay.
01:32
So in looking at the graph, we got to look at kind of what's happening.
01:38
See here, we're kind of increasing, and then at this point, we stop increasing, and we start decreasing.
01:48
Okay.
01:49
Now, what that means is our, remember, this is our distance.
01:57
This is time.
01:59
So what this is telling us is here, this first little part during that first little interval of time, our distance is increasing.
02:09
We're getting farther and farther away from the origin.
02:14
Okay? but then at a certain point, we're not, we like kind of max out here.
02:22
We max out.
02:23
It says zero right there.
02:25
We're not getting farther away.
02:27
We're not getting closer.
02:28
But then we start coming down.
02:33
We start coming down.
02:35
Which means we're getting our distance is getting closer to the origin, which means change direction is coming back.
02:49
So this point here, or the vertex, is the time when it's going to change direction.
03:04
So where is it? so the value of q is going to be 4.
03:11
So q equals 4.
03:13
Now, i don't know how much you've done with derivatives, but if we take the derivative of s, we're going to get 8 minus 2t, and when you set that equal to 0, negative 8 equals negative 2t, t equals 4, there it is.
03:36
This is the velocity.
03:41
Okay, so before we hit time equals 3, 4, our velocity was increasing, it was positive.
03:50
But then after we had to time equals 4, our velocity hits zero, and then it starts to, it becomes negative, which means we have velocity, but it's going in the opposite direction.
04:03
It's going into the left.
04:05
Okay...