The position of a particle P on a line / is given by s(t) =...

The position of a particle P on a line / is given by s(t) = 2t^3 - 4t^2 + 2t - 1, where s(t) is in feet and t is in seconds. Find the velocity as a function of time. v(t) = Determine the time intervals (in seconds) in which the particle moves in the positive direction and the negative direction. (Enter your answers as comma-separated lists of intervals.) positive direction negative direction At what time(s) (in seconds) is the particle at rest? (Enter your answers as a comma-separated list.) t = s The position of an object that is projected straight up from the ground with an initial velocity of 116 ft/sec is given by s(t) = 116t - 16t^2. What is the velocity of the object (in feet per second) at times t = 2, t = 3, and t = 4? v(2) = ft/sec v(3) = ft/sec v(4) = ft/sec At what time (in seconds) does the object reach its maximum height? (Round your answer to one decimal place.) t = s At what time (in seconds) does the object strike the ground? t = s What is the object's velocity (in feet per second) when it strikes the ground? v = ft/sec

Transcript

00:01 Okay, so the thing to know is that if this is the position function for time t, we get st is equal to 2t cubed minus 4t squared plus 2t minus 1.

00:26 And so the thing to know is that the velocity function is the derivative of the position function.

00:32 So we're going to take the derivative of that equation to get, by using the power rule, we get 2 times 3 t squared minus 4 times 2 t plus 2 and then then minus 1 goes away because it's a constant when we take the derivative of a constant it's 0.

00:54 So the velocity is given by 6t squared minus 8 t plus 2.

01:07 And so now to figure out where it's positive and negative, we need to divide up the number line among where the velocity.

01:15 Is zero.

01:16 So we're going to do actually the fourth problem, the fourth box we're going to do before we do the second and third box because we need to figure out where the velocity is zero.

01:26 So when vt equals 6t squared minus 8t plus 2 equals zero, first i'm noticing i can divide both sides of this.

01:45 I can divide both sides of this by two and get three, t squared minus 4 t plus 1 equals 0 and i'm going to use a quadratic formula calculator to calculate this i go to google and type in quadratic formula calculator if you go to this one calculator soup .com you get you can enter the coefficients of any quadratic and it'll tell you the roots so i'm going to enter 3 negative 4 and 1 3 negative 4 and 1.

02:32 Calculate.

02:35 And we get 1 3rd and 1 as our answer.

02:42 There are two roots, these 2, 1 3rd and 1.

02:46 So at time 1 3rd, the velocity is 0.

02:52 And at time 1, the velocity is 0.

02:57 So now i'm going to divide up the number line among those zeros.

03:01 What i mean by that is i'm going to start at time 0 and go to time 1 3rd, when the velocity is zero.

03:09 Then i'm going to go from one -third to one, divide the number line between where the velocity is zero.

03:16 Then i'm going to go to one to infinity.

03:26 All right.

03:27 So we take our function, 3t squared minus 4t plus 1, and we evaluated it at a test point in each of these intervals.

03:39 So i'm going to evaluate this at 1 fourth, one -half, and 10.

03:54 So those three are values in those intervals.

03:57 So i'm going to plug in one -fourth for t and see if it gives me a positive or negative result.

04:02 If it gives me a positive result, then you can tell that the particle is moving in the positive direction.

04:09 If it gives me a negative result, it's moving in the negative direction.

04:13 So let's go to desmos .com.

04:19 Go to the math tools and go to scientific calculator.

04:22 And let's use our tools to make this easier on ourselves.

04:26 So i'm going to type in 3 and we use some parentheses as a placeholder for this t minus 4t.

04:36 That should be squared by the way.

04:38 Squared.

04:39 There we go.

04:40 Minus 4t plus 1.

04:45 Minus 4t and again i'm going to replace key with parentheses and i want to plug in 1 4th first and i get a positive number.

05:02 I don't care what the value of the number is.

05:04 I just care about its sign and sign is positives.

05:08 So we're going to put a plus here.

05:10 That means it's moving in the positive direction from 0 to 1 3rd.

05:18 From 0 to 1 3rd, it's moving in the positive direction.

05:21 Now let's check 1 half and see what the sign is at 1 half.

05:31 We get negative 0 .25.

05:33 So it's negative.

05:35 So on one -third to one, when time is between one -third and one, we're moving in the negative direction.

05:45 And finally, let's plug in 10.

05:53 We get a positive number.

05:57 So there on one to infinity, it's moving in the positive direction again.

06:02 So you can say your positive direction, you'll list the intervals 0 to 1 -3rd and 1 to infinity.

06:24 And your negative direction, you'll list at...

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