Practice: A car accelerates from rest at a constant 3 m/s^2 for 15 s, after which it begins to decelerate at a constant rate. The car comes to a complete stop after decelerating for 200 m. Calculate the car?s average velocity from start to end.
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Concept #1: Motion with Multiple Parts
Hey guys so a lot of motion problems are going to have multiple parts and what I want to do in this video is show you two examples as well as some techniques to make this more manageable so let's get started. So like I said multiple parts what we want to do is draw a complete interval diagram and write one equation for each interval for example I'm going from A to B and then from B to C. So how many intervals are here some of you might be thinking 2 but it's actually 3 there's this one B to C and then there's a larger interval A to C and the key thing here is that you need to write one equation for each but when you write those equations make sure that the variables you are using correspond to that interval for example let's write an equation for this interval here let's say that the average velocity between B and C I'm sorry A and B is the equation for average velocity is delta X over delta T but if I'm doing this for A and B I have to make sure that I use delta X for A and B and delta T for A and B. So that's one thing the other thing I want to talk about is the idea that if I'm going from A to B this is my initial and this is my final. But if I'm going from B to C, B is my initial and C is my final So this presents a potential problem because B is final velocity for of the first interval and initial velocity for the second interval and that's not a problem in itself the problem is that if you have a long question with a lot of different variables you might get confused you might get caught up trying to remember that final velocity one is the initial velocity of the other so to avoid that what I like to do is lets say the speed at A I just call that V.A I just call this V.B and I call this B.C because guess what V.B is V.B for the first half of the second half it's the same number and it's the same letter and I think that makes it easier so let's do a problem a car travels with a constant 50 so constant velocity acceleration 0 for 10 seconds and then a constant 30 acceleration is 0 for 600 meters let's draw the complete interval diagram. I have two intervals so I do this technically there's a third larger interval there A B and C now I'm going to put all the information I have for these points so here it says I travel with a constant velocity so I can call this velocity between A and B it's an average velocity of 50 and since I'm putting that for the entire interval you should know that that's a constant velocity or average velocity for the whole interval not the specific velocity at A or B but sort of in between the acceleration for A and B is zero what else do I know I know the time between A and B is 10 alright for the second part I know that I know that the velocity between B and C is 30 so the acceleration between B and C is 0 and the delta X for B. and C is 600. Let me put units here meters per second these are all standard units so we don't really have to worry about units.
Cool so that's the interval diagram are you going to get a question to ask you to draw an interval diagram? No that's just a way for you to visualize how this stuff works which should always be drawn in the situation anyway so we got that what is the total distance traveled total distance now notice that I have delta X's that's displacement but the questions asking for distance guess what if you think that you are just moving to the right which you are so think about it in terms of you just moving to the right so it doesn't really matter the diff the difference between distance and displacement here doesn't really matter so I'm just going to think of this as my delta X. Now it's the total distance so it's delta X from A to C and all you got to do is piece them together if you move 10 and then you move 20 then obviously your delta X is just 30 so this is just the sum of delta X ray B and B, C. In fact the same thing happens with time the total time from A to C is just the time from A to B plus the time from B to C. Alright so if I want my total delta X I need to have those two other numbers I don't know delta X A I don't have this sad face but I have delta X, B, C that's 600. So to get this answer let me go ahead and do this kind of organizing the work here to get this answer I'm going to need to get delta X A B. So let's go do that delta X A B belongs to the first interval, so how do I find a number in physics you write an equation you solve the equation I mean the first interval here A to B so I'm going to write an equation for this piece right the acceleration is 0 so there's only one equation you can write which is that the average velocity is delta X over delta T. So if you're solving for delta X then you just have the delta X is V, T, notice I didn't write delta T I just wrote a T. I didn't write V average I just wrote V and that's fine you don't have to write all the little letters every single time. So to find delta X A B, here I'm going to be a little bit more precise I'm going to write V A B and T A B and I do have those two numbers I do have those two numbers the velocity is 50 and the time is 10. So my delta X is just 500 meters. Now I can plug this here and my total answer is 1100 and that is part B, A and B are done. Now lets do C what is the average velocity from start to end remember the definition of average velocity, definition of average velocity is delta X over delta T but here I want for the total motion from A to C so I need delta X from A to C and delta T from A to C. I know my total delta X it's 1100.
But I need my total time I don't have the total time once I know the total time I have the final answer for this so let's go find our total time. To find a total time total time is right here I know the first time is 10 seconds but I need the second time I need the second time so let's find the time between B and C. How do you find a number in physics you find an question so delta T. B C belongs here so I have to write an equation for that interval and the only equation I can write again is that the average velocity is delta X over Delta T. Here I'm looking for time so if I move things around I find that delta T is V over X. So if I want to find this delta T. B C is just the velocity between B and C and the delta X between B and C and the velocity between B and C, did I get that right nope I got it backwards it's actually X over V, X. over V wops big mistake. Delta X over V B C cool caught that on time. So delta X is going to be 600 and the velocity here is 30. So this is 20 seconds. So my second time is 20 seconds so my total time from A to C is just the addition of my two times and it is going to be 10 plus 20 equals 30. So I plug in 30 here divide 100 by 30 and you get I have this here 36.7 and that is the answer to part C. So you work on each interval separately and then you just piece everything together to find the total average velocity. Alright so let's do the next one now, I'm going to solve this one as well the first one had no acceleration this one will have acceleration but they're both multiple parts alright. So here it says you're driving at 30 meters per second when you see a traffic light turn yellow so you immediately hit the brakes causing the car to decelerate. This is a classic problem of reaction time and the idea of reaction time is when you see something and you immediately react on it it's not actually immediate It takes a little bit of time which is usually 0.7 for you to do something so that's the time that it takes for your brain to process information for you to react to it and the idea in these questions is that for those 0.7 you don't do anything right you're actually just moving at the same speed and then you start decelerating, so it's kind of dangerous but let's see. You're driving at 30 when you see so the idea that the first interval that first 0.7 seconds this first 0.7 seconds. You don't actually do anything right you're still sort of reacting to it therefore this is a two part problem A to B is your reaction and this is where stuff is actually happening right A to B. to C. So you cause the car to decelerate at a constant 7 that's for the second piece the first piece you do nothing actually right you're just like reacting to it so the acceleration for this piece is 0 the acceleration for this piece is 7 but it's negative. Now instead of writing A B and B C that could get kind of annoying I could just call this interval 1 and I could call this interval 2,if I would like. So I can do T1 and A1 and then this is A2, so the velocity here at point A is 30 and we want to cause the car to stop and to break until it stops so I want V C to be 30 as well now I notice that my acceleration is 0 for the first part that means that my V.B is, actually this is zero sorry. That means my V.B. will be 30 as well because between A and B you're just reacting to it you're acceleration is 0 your velocity doesn't change right you keep moving at the same rate here you're going to come to a stop eventually.
So let's see what it says here first question asks us to find how long you will travel in the 0.7 seconds that it took you to react so basically between A and B. I want to know what is your delta X? That's what that's asking what is your delta X from A to B? Alright so obviously A to B belong to the first interval, A to B is the first interval so I'm going to write an equation for that first interval and what equation can I write the acceleration is 0 so the only equation that's relevant here is that the average velocity is delta X over delta T and I'm looking for delta X, so that's what I'm going to do I'm going to solve for delta X here move stuff around so it's V T for that interval the velocity is 30 the time is 0.7 so if you multiply this you get 21 meters so you cover 21 meters in the time that it took you to even react, we got that. Then it says once you start breaking how long in seconds does it take for you to stop? So once you start breaking how long does it take you to stop? Once you start breaking means B. So how long does it take you to stop from there forward is the time from B to C. So part B is asking us what is the delta T for that second interval? And how do you find the number? You write an equation? So I have to pick an equation for this second part the acceleration is not 0 so I can have as many as 4 equations to pick from right and they are up here. Hoops the top of the page, now remember your professor may or may not allow you to use that fourth equation. Hopefully we don't even need it anyway, so once you start breaking how long does it take? let's list what I have right V initial in this case is V initial is your V B and its 30, V final is your V C and it is 0 I wrote it 20 here not sure why, the acceleration is -7 and what else do I know delta T is what we're looking for I already have 3 and so delta X is the fifth variable is the ignored variable the one that I don't want I don't know I just don't care about the delta X variable being the ignored one tells us that I should use the first equation to solve this because the first equation doesn't have delta X remember that right so V equals V initial plus A T and the final velocity is 0 the initial velocity is 30 the acceleration is -7 time so I can solve this by leaving time alone I can move the 30 to the left so it becomes negative and I divide both sides by 7 and I get that time is I have it here 4.29. So my time is 4.29 seconds and that's the answer that we were looking for, so A B that's the time there. Now Part C. is going to ask what is your total stopping distance so again I'm just moving to the right you can in one direction you can just think of distances displacement there is no difference between them in this one so I can just say what is my delta X from A to C and obviously the delta X from A to C is going to be A B plus B C, do I know A B ? Yes I know A B but I need B C so this is going to be delta X, A C is going to be the first one plus the second one I know the first one 21 meters but I need this guy here and once I have that guy I know the entire answer. So now let's find delta X B C and guess what delta X B.C is the only variable that I still don't know for that interval I actually at this point I know 1,2,3 and I know the time now is 4.29 I know four out of five so it should be pretty easy to solve this when you know four out of five you just need three out of five but if you know four you actually have two equations they're going to be able to pick from, so hopefully we can pick the easiest one and I'm just going to go with I'm just going to go with the third equation just because it's kind of an arbitrary selection but its just because it starts with delta X equals you don't have to move things around. So delta X. equals V initial T half A T squared so now all I got to do is plug in these numbers the initial velocity here is the velocity at B. which is 30. The time that it takes between B and C we already found it is 4.29. This is half the acceleration is -7 and then the time is 4.29 squared and when you do all this I get 64.3 meters is my delta X B to C. But we're not done yet you got to plug this in here. So 64.3 so my final answer is 85.3 and that is the total displacement from A to C. Right so once again in these questions you are going to break them into small pieces and then put it back together at the end. Alright that's it for this one.
Concept #2: Given/Target Problems
Hey guys, so a very typical type of problem in physics is what I call the given target problems and these are going to show up not just in this chapter but pretty much in almost every chapter from now on. So let's see how these work. So the basic idea is that you will give they will give you information about a certain situation and then ask you about a different but related case. So there's going to be two similar situations and what we're going to have to do is solve for a variable in the given situation that we can use in the target situation and if you can identify these questions then solving them becomes a little bit easier. So let's see, I'm going to do one example and then I want you guys to try practice one. So a car starting from zero, starting from zero initial velocity equals zero, reaches 40 meters per second in just 8 seconds so let me draw an interval diagram here. So you reach 40 meters per second that's your final velocity obviously there's an acceleration in just 8 seconds. So this is my given situation, I'm giving you this information and then later I ask how far in meters, so delta X, will it have traveled after 6 seconds? So I'm telling you that in 8 seconds it reaches this velocity and then I'm asking you for the same car for the same situation how far will it be after 6 seconds? So if this is 8 seconds then obviously 6 seconds would be a little bit before that point so I'm going to say that this is a delta T of 6 seconds. It's the same car, it's the same motion so this V initial here is still zero and I want to know the delta X for this situation here. So if you want to solve this one way you can go about it you can go straight for the answer, try to go straight for the answer and find this delta X and that's what I want to do because in a test that's probably what you would do rather than have an entire strategy already you should probably just go over delta X and if you get stuck you go find some other variables. So without thinking about it too much I'm going to go for delta X and the way I solve motion problem just by listing my five variables, so V initial, V final, A, delta X, delta T. Delta X is what I'm looking for, delta T is 6 seconds and the initial velocity is zero. Remember to solve any of these questions I need to know three out of five variables, I only know two. Delta X is my target so the third variable is going to have to be either V or A. That V is the final velocity and now this is where I'm going to have to go to the given part of the problem and to find one of those two variables that I can now use on the target part of the problem. So V or A are going to come from up here and you have to figure out which one. So first let's look at V. V isn't going to be the one that I can borrow from the given because they have different V's. If this guy's been accelerating for 8 seconds it's obviously already moving faster than this guy that's accelerating for 6 seconds so I'm going to say that V, I'm going to call this one and two, I'm going to say that V final for one is not the same as V final for two however it's the same car, the same motion, the accelerations will be the same and that's what I'm going to borrow from the first part so I can plug it into the second part. So I'm going to go over here and say hey I need to know the acceleration and the idea of the given part of the problem is that you are given more information so you can find something that then you can use on the other one. So we should be able to find the acceleration here. Let's go here and try to find the acceleration for this first problem. How do I find a variable in a motion problem? I list my my five variables, my kids here and I try to find the right equation and solve. V initial, V final, A, delta X, delta T. Obviously these are all coming from the given part, so I'm going to be looking at these numbers not the numbers for the second part. So the initial velocity is zero, I know the final velocity is 40, I know this takes a time of 8, I'm looking for the acceleration. I have three out of five variables so I can solve. Notice that delta X is not given and not asked for. Delta X is my ignored variable and now I know which equation I can use, I can use the equation that doesn't have delta X to find A. That is the first equation according to how I list them. So V final is V initial plus AT. The final velocity is 40, the initial velocity is zero, the acceleration is what I'm looking for and time is 8. So acceleration is 40 divided by 8. The acceleration is 5.
And now that I know the acceleration is 5 I can come back here and say acceleration is 5 right, so now I finally have three out of five variables and I can solve for the time. So now I can solve for for delta X rather. So how do I solve for delta X? Well if I have the acceleration, the final velocity is now my ignored variable, the final velocity is now my ignored variable and that means that I would be using the third equation to figure this out and the third equation says delta X equals V initial T but the initial velocity is zero, plus half of AT squared and I know that the acceleration is 5 and the time again I'm on the target situation now so the time is 6 not 8 so this is 6 squared and if I plug all of this in, do I have this here, I get a 90 meters. So the final answer is 90 meters. So again remember the process, you're going to find some information in the given part and you have to make sure that that information is applicable to both and shared between both and then you're going to use that information in the target part to figure out what you need. So again these things are going to show up a lot. Alright, so now I want you guys to give practice one a shot. It's very similar you should be able to get it and I'm going to jump right into it and show you how this works. Suppose it takes you 10 seconds to reach 50 meters per second from rest let's draw that. So it takes you 10 seconds to go from rest V initial equals zero to over here a V final of 50 meters per second and the time between these two is 10. Then it says accelerating at a constant rate, cool, acceleration is always constant. What is your speed after the first 100 meters? So this is my given situation and this is my target situation where I want to know for this motion what is the speed after 100 meters? Here's the problem with this, I don't know if 100 meters happens I can't know right away if 100 meters happens somewhere before I get to 50 meters per second or somewhere after I get to 50 meters per second I just don't know that. So when I draw these two diagrams I won't be able to sort of visually compare the two. So what I'm going to do is just kind of draw them same length but I'm not going to make a little connecting line here that's just the way you organize information there's no proper way that you're supposed to do this it's just for your own sake right but it's the same situation so I do start from the same initial velocity of zero so that part is the same and here my delta X equals 100 and I want to know what is your speed here? So these are not the same, I'm going to do a little not sort of vertical not the same which means that the times are not going to be the same either necessarily. So let's jump right into trying to find the answer. How do I find a variable in a motion problem? I list my kids and I get the right equation so let me list them here, make sure I don't miss anything. The initial velocity is zero, the final velocity is what I'm looking for, the acceleration here I don't have it, delta X is 100 and delta T I don't have either. So the inventory I got two variables, I need three so the third one is going to be not V because V is what I'm looking for but it's going to be either A or T and which one do you think it is? I hope you're thinking that it was A that's the same. It's the same motion just different pieces of it so it's a constant acceleration so A is the same. T cannot be the same because if T was exactly the same in these two situations they are really different and then that just means they're the same interval that just means your velocity would be 50. So T, I'm going to say here that T1 is not the same as T2. Now it could have been the same if they happened to be the same. If it happens that it takes 100 to get to 50 meters per second, those numbers will happen to be the same but that's usually not the case. The acceleration however will certainly be the same. So the accelerations are the same which means I have to find the acceleration for this piece so that I can then plug it into the bottom. So let's find the acceleration for the first half. V initial, V final, A, delta X, delta T. This is for the first problem, the given part of the problem. The initial velocity is zero, the time is 10, the final velocity is 50 and I'm looking for the acceleration. Notice how on the given problem I always have enough information to find something else. Here my ignored variable is delta X, sad face, and that tells me I should use the equation that doesn't have delta X which happens to be equation number one, the easiest one. V equals V initial plus AT and if I'm looking for A I just have to plug everything else. The final velocity's 50, the initial velocity is zero, I'm looking for A and the time is 10 so if you move things around acceleration is 5. Another way that you could have done this is just understanding that the acceleration is a change in velocity over a change in time by the way these two equations are the same it's just different ways of writing it and that would have given you 5 as well. Now I can use this 5 over here, now I know that A is actually 5 that's coming from here boom and now that I know that I can find V final. So let's find V final for this piece. Notice that now that I have three out of five, my ignored variables time, sad face time, which means I'm going to use the equation that doesn't have time the equation that doesn't have time is the second equation. V squared, V initial squared, 2A delta X and I'm looking for the final velocity this guy right here. The initial velocity is zero so the final velocity is the square root of 2, A is 5, delta X is 100 if you do all of this I have it here it's 31.6 meters per second. That is your final velocity. So once again remember the process you find some information the first then you apply to the second. Let's keep going and I'm going to do one more example for these types of problems. In a 100 meter race, you accelerate from rest at a constant 3 meters per second. So the race is 100 meters long, so it's got a delta X of 100 meters. You accelerate from rest, V initial equals zero and you have a constant acceleration of 3. Your friend I'm going to write F he also accelerates from rest. It doesn't really say that but it says your friend accelerates 2 meters per second obviously starts from rest and the question is how far into the race is your friend when you cross the finish line? So this is similar but instead of a given situation and a target situation it's now comparing two different people but it's the same exact process and you want to know for your friend we want to know what is his X final or what is his total delta X when by the time you cross the line. So by the time you finish. So the delta T of your friend at this point is your delta T of crossing the line. That's kind of ugly but the idea here is that you can't figure out his final delta X until you know how long it takes for you to finish the race. Let me show you that. Let's do what I've been doing which is jump straight to the answer and try to find it and you see we'll get stuck but then we'll figure out how to get out if. Alright so V initial equals zero, V final we don't know we don't care, the acceleration of your friend is 2, his delta X is what we're looking for and delta T we don't have it either. So again I have two things, I need a third one and what are we going to get? Well this problem says how far is he when you cross the finish line? So it's sort of tying his time to your time. At that time, how far is he? So it's basically saying we want his time to be your time when you cross. So that's the variable that's sort of common to both so I'm going to have to go to you and get the time. So we're stuck here for now, let's go here and figure out what is your delta T. Again, I list my five. V initial, V, A, delta X, delta T. The initial velocity, your initial velocity is zero, your final velocity we don't know, your acceleration is a 3, your delta X if you're finishing your delta X is 100 and time is what we're looking for. Notice how I have everything that I need, three out of five. My ignored variable is the final velocity and that tells me that I should be using the third equation to be using the third equation so delta X equals V initial T but it's zero plus half of AT squared and out of all of this I'm looking for time. So let's get time out of there by itself so I'm going to move the 2 over here and I'm going to move the A down here. I still have time squared on the other side so I can do this. Now I just got to plug in all these numbers. This is the square root of 2, 100 divided by 3 and I did this in a calculator earlier it's 8.16 seconds. So 8.16 seconds is your time which means his time is going to be 8.16 seconds as well which means I now have three out of five variables and I can solve for delta X. When solving for delta X we're going to notice that V is your ignored variable, sad face, so the equation I'm going to use once again is the third equation. So I can use the third equation here and the third equation says that delta X equals V initial T plus half of AT squared, this is zero, and we want to know delta X so all I got to do is plug in these numbers. Half the acceleration is 2 and the time is 8.16 squared. You plug all the stuff in here and you find out that his delta X is 66.7 meters. One other way that this question could have been asked just to kind of wrap up here they could have asked what is the by the time you cross the finish line what is the gap between you two? And obviously the distance between you two I'm going to call this distance between you and your friend by the time across the finish line would have been just 100 which is yours minus 66.7 which is his distance and that would have been just 33.3 so that's another way you could have been asked that which would have required sort of an extra step. When you put this in parentheses because you weren't really supposed to do this just wanted to talk about. Anyway, that's it for this one.
Concept #3: Motion with Multiple Parts (Practice Intro)
Practice: A car accelerates from rest at a constant 3 m/s^2 for 15 s, after which it begins to decelerate at a constant rate. The car comes to a complete stop after decelerating for 200 m. Calculate the car?s average velocity from start to end.
Practice: A car decelerating at 5 m/s^2 stops in 120 m. How much does it travel in the first 2 s after it starts braking?
Practice: You and your friend are driving on the highway with constant speeds of 50 m/s and 40 m/s, respectively. Assuming you were originally behind, how far from him/her will you be 5 seconds after passing?
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