00:01
In this problem, we are given the position of a particle that is s of t equal to t raised with third power minus 9 t squared plus 24 t.
00:11
Now for the first part, we want to find the velocity, which is just the first derivative of s.
00:19
And that's equal to 3t squared minus 18t plus 24.
00:25
And then for the value of the velocity, when t equals, one we simply plug it in our equation in a and that's going to be three times a square of 1 minus 18 times 1 plus 24 this gives us a value equal to 9 feet per second as you want to find the time when the particle is at rest now a particle is at rest when the velocity is zero since you already have the velocity function in part a we have 3 t squared minus 18 times t plus 24 that's equal to 0 can factor out the 3 we have t squared minus 6 times t plus 8 equals 0 this gives us 3 times t minus 4 times t minus 2 equal 0 which gives us t equals 4 and 2 so the time wherein the particle is at rest will be at 2 seconds and at 2 after four seconds.
01:34
Next you want to find the time when the particle is moving in the positive direction.
01:40
So i want to say particle moving in the positive direction, this is when the velocity is greater than zero.
01:48
So since you already have our boundaries, that will be the values of t when v of t equals 0, we will then create a number line and partition this.
02:02
Partition this using the values of t we have from c so our intervals would be from negative infinity to two two to four and then four to positive infinity next we want to find the sign of v of t in each interval and to do that we have to pick values inside each interval so let's say for this we pick zero for this we pick three and then for this you pick five so when a t is zero we have v of t which is equal to 24, so this is positive.
02:38
When t equals 3, we would have a negative value, and when t equals 5, we would get a positive value for v of t.
02:51
And since t is time, then it must be starting at 0.
02:58
And so the interval in which the particle is moving the positive direction would be from 0 to 2 and 4 to infinity.
03:09
And then for the next part, we have to draw a diagram on the movement of the particle.
03:18
And then from here we would calculate the distance traveled by the particle.
03:22
Using the information we have from e, you would say that starting at t equals 0, and that would be s equals 0.
03:32
Since if we plug it in our s, we will get a value.
03:35
Equals 0 going to 2 that's a positive direction so i must go to the right so t equals 2 the value of s here would be 20 now in between 2 to 4 the sign of v is negative so it goes to the left until t equals 4 and when t is 4 we have s that's equal to 16 and then after t equals 4, it goes back to the right, since we are now in the positive direction, for 4 to infinity.
04:30
And then it stops at t equals 6.
04:34
And at t equals 6, s must be 36.
04:38
And so the distance traveled from t equals 0 to t equals 6, that will be the distance traveled from 0 to 2, plus a distance traveled from 2 to 4, plus a distance traveled from 4 to 6.
05:00
So from 0 to 2, that will be a distance of 20, plus from 2 to 4, that will be 20 minus 16 since we went to the negative direction, plus from 4 to 6, that will be a positive direction.
05:24
So it's going to be a 36.
05:27
6, we have to subtract the 16, which came from the distance travel that t equals 4.
05:39
And this will give us a value equal to 44.
05:43
And then we are asked of the acceleration of the particle.
05:47
That's a of t, which is just the derivative of v.
05:52
And this is equal to 6 times t minus 18.
05:58
And then we are asked to find the value of the acceleration when t equals 1 and that's going to be 6 times 1 minus 18 or negative 12 feet per second squared...