A particle moves according to a law of motion s=f(t), t ⩾0,...

Key Concepts

Speeding Up versus Slowing Down

Determining whether a particle is speeding up or slowing down involves comparing the signs of the velocity and acceleration functions. When both have the same sign, the speed increases; when their signs differ, the speed decreases. This analysis is essential in understanding the dynamic behavior of moving objects and shows the interplay between velocity and acceleration.

Graphical Representation in Kinematics

Graphing the position, velocity, and acceleration functions provides a visual summary of motion. These representations help in understanding the relationships between the functions, interpreting critical points, and visually identifying intervals where the object speeds up or slows down. Such diagrams are valuable tools for both qualitative and quantitative analysis of motion.

Total Distance Traveled

Total distance traveled is the accumulation of all movement regardless of direction. This is found by integrating the absolute value of the velocity function over the given time interval. This concept is fundamental in distinguishing between displacement (which considers direction) and total distance (which does not).

Quotient Rule

The quotient rule is a technique used in differentiation when the function is expressed as a ratio of two differentiable functions. It is particularly relevant in this context because the position function is given as a quotient. Mastery of the quotient rule is essential for correctly computing both velocity and acceleration from the position function.

Derivative as Rate of Change

In kinematics, the instantaneous rate of change of position with respect to time is called velocity, which is found by taking the derivative of the position function. Similarly, acceleration is the derivative of the velocity function. This concept forms the basis for analyzing how an object’s position changes over time and how quickly these changes occur.

Sign Analysis of Functions

Analyzing the sign (positive or negative) of functions, such as velocity and acceleration, is crucial for understanding the direction of motion. Determining when the velocity is positive, negative, or zero allows one to identify when the particle is moving forward, backward, or at rest. This technique is a key component in the study of motion and dynamics.

Transcript

00:03 A particle moves according to a log motion.

00:06 S equals f of t for t is gray linear equal to zero, where t is measured in seconds and s in meters.

00:13 Find the velocity at time t for part a.

00:18 Okay, so to get velocity, velocity is the derivative of the position function.

00:25 And so to do the derivative of a fraction, we use the quotient rule, which i have written in red, where the top is going to be your u, the bottom is going to be your v.

00:39 And so you're going to do u prime, so that would be the derivative of 9t, which would be 9, times v, v is the bottom, so the t squared plus 9, minus u, so minus 9t, times the derivative of v, the derivative of t squared plus 9 would be 2t.

01:12 And then that's it going to be over v squared.

01:21 So t squared plus 9 or v squared.

01:38 And then i did a little moving over quotient row so that there's space to work here.

01:47 Okay, so if you take and multiply the 9, and you're going to get 9t squared plus 81 minus 18 t squared over t squared plus 9 squared which is going to equal 9 t squared minus 18 t squared would be minus 9 t squared plus 81 over t squared plus 9 squared okay then on part b it says what is the velocity after one second so the velocity after one second all you have to do is put 1 into that and so it would be minus 9 times t squared would be 1 squared, so times 1 plus 81 over t squared, which would be 1 plus 9 squared.

03:19 So we'd have 81 minus 9, which would be 72 over 10 squared, which would be 100.

03:28 And so that would be equal to 0 .72, in this case, meters per second.

03:54 And part c says, when is the particle at rest? particle will be at rest when your velocity is equal to zero.

04:07 So your velocity function is what we have up above.

04:40 Okay, and one of the things i want to do is, i go ahead and factor out a negative 9 on the top of that.

04:54 And the reason i did that is because to find out where this equation is equal to zero, you need your top to be zero and your bottom to not.

05:07 And so we're going to want to know what values the top is equal to zero, but we also need to see whether or not any of our factors will cancel out from top to bottom.

05:20 And t squared plus nine does not factor.

05:22 And so that part won't simplify, but our top will simplify one more.

05:32 T squared would be t plus 3 and t minus 3 is what that would factor into.

05:42 And so nothing will cancel out between the top and bottom, but what we want to do is we want to set just this top portion equal to zero because that's where the whole fraction will be zero.

05:58 And so by doing that, you would say t plus 3 equals 0, so t equals negative 3, and t minus 3 equals 0, so t equals 3.

06:16 Now remember, our t is supposed to be greater than or equal to 0.

06:26 And so we're not going to use this one.

06:30 So we just have when is a particle at rest, it'll be rest for 2.

06:35 T equals 3.

06:37 Sorry, that kind of covered up part of the answer.

07:03 And that would be seconds.

07:08 Okay, part d says when is the particle moving in the positive direction? so to determine when it is moving in the positive direction, you need the velocity to be greater than zero.

07:35 And so for our velocity function, if you take a look at the velocity function, the bottom part of this is always going to be positive.

07:51 Because a t squared, anything squared plus 9 would be positive, or if you just think about the entire thing being squared, anything squared would be positive.

07:59 So that part of it's always going to be positive.

08:03 So for the overall function to be positive, then we need the top of that fraction to be positive also.

08:13 Okay, so we're going to take and set the top part of that to be greater than zero, in other words, positive.

08:26 And then if we look at dividing through by a negative 9, when you divide through by a negative number, it is going to flip your sign.

08:52 And so you're going to have t plus 3 times t minus 3 needs to be less than zero.

09:07 Okay, so that means t plus 3, if you think about as a less than you can, you can say t is going to be less than negative 3, or t would be less than 3.

09:34 The thing about inequalities like this is you still need to set them up essentially on a number line and figure out where it's going to work.

09:47 And so you're breaking at negative 3, which negative 3 we don't really want to worry about because we're going to.

09:54 Talking about zero and up.

09:57 So if you look at zero and up, then you're breaking at three, and you need to check values in each of those sections to see where the statement will be true.

10:08 And so an easy one to check, since it does include the endpoint, you can use zero in there.

10:17 So if you do zero into this right here, then you'd have negative nine times zero plus three, would be 3 times 0 minus 3 would be a negative 3.

10:33 And the question is, is that going to be greater than 0? well, a negative 9 times a negative 3 would make a positive 27 times 3.

10:44 So definitely going to be a positive number or greater than 0.

10:56 You don't really have to know what the value is.

11:00 You just need to know it's positive.

11:02 So in here, this is positive.

11:05 And if you want to check one, you can check one.

11:09 Zero, it's okay to check it because t needed to be greater than or equal to zero.

11:14 If it said t was greater than zero, only you couldn't use zero.

11:20 Okay, and then we want to check three, something in this next interval, so we want to use something bigger than three, so such as four.

11:32 So this one was for t equals zero.

11:37 This one's for t equals four.

11:42 So if you do negative nine times the first one will be four, plus three, which would be seven, times four minus three, which would be one.

11:52 You're going to get a negative 63.

11:55 So in this interval, it's going to be negative.

12:00 And so since the velocity is greater than zero, it's going to be moving in a positive direction.

12:07 Then in this case, it's going to be moving in a positive direction from zero up to, but not including three.

12:17 Because at three, it would be equal to zero, and it would be at rest.

12:20 Our e says find the total distance traveled during the first six seconds.

12:35 In order to do that, we're going to need our position function, or we can say f at t, k.

13:08 And then since the particle is moving in the positive direction from 0 up to 3, and then after 3 it's moving in the negative direction, then it's moving in different directions.

13:26 And so we need to break this interval up from 0 to 3.

13:35 Well, really, we're going to go ahead and do the end point for distance.

13:42 And so we're going to break it up from 0 to 3 and then 3 to 6 because it said up for the first 6 seconds.

13:58 And we're going to take and do f at the absolute value of f at 6 minus f at 3.

14:11 Actually, i'm going to do the 3 and 0 first.

14:15 Maybe f at 3 minus f at 0 and that'll give us the distance from the point x equals or t equals 0 to t equals 3 and then we'll add to that the distance between f at 6 and if at 3 okay so if you take and plug in 3 into your equation over here then 9 times 3 would be 27 over 3 squared would be 9 plus 9 would be 18.

14:59 And then minus f at 0 would be 0.

15:15 So if you want, you can get a decimal for that.

15:26 Hopefully it's not repeating me.

15:30 Yeah, that's one and a half.

15:32 That wasn't even weird.

15:36 Okay, and then if you do f at 6, 9 times 6 would be 54 over 6 squared would be 36 plus 9.

15:58 So that would be 54 over 45.

16:12 And then you're going to subtract off f at 3.

16:15 Well, f of 3 was this right here.

16:18 So minus 1 .5 or 27 over 18.

16:24 So you end up with, and you want the absolute value of this.

16:31 So you're going to end up with, for this one, the absolute value of 1 .2 minus 1 .5.

16:40 Which would be a negative 0 .3, but essentially with the absolute value would be 0 .3.

16:46 So then you add those two together.

16:47 Together, 1 .5 plus 0 .3 is going to equal 1 .8 meters.

17:08 Part f says draw diagram, like figure 2 to illustrate the motion of the particle.

17:20 So between 0 and 3, particles moving into positive direction.

17:31 So if you kind of think about having the number line where you're at 0, you're going to go up to 3 for your t.

17:48 Then starting out at zero here, you're at t equals zero, and your s is equal to zero, or your f of t.

18:05 And it's gonna move in the positive direction until it gets to three here.

18:22 Okay, and you want to mark that direction on there by using arrows.

18:32 And then from three to six, it's gonna be moving in the negative direction.

18:44 So you're going to turn and go the opposite direction.

18:47 Now the thing is, from 0 to 3, it went 1 .5, and from 3 to 6, it's only going to go 0 .3...

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