Live from Las Vegas, it's theCUBE, covering AWS re-invent 2018.  Hey, welcome back everybody. Jeff Frick here with theCUBE. We're at AWS re-invent 2018 in Las Vegas. Day two of four days of coverage. I think we'll do 120 interviews. I mean, this is the most hoppin' show in tech right now. We're really excited to be here. And joined by my co-host Lauren Cooney, Lauren, great to see you. Thank you, great to see you too. We've got our next guest. It's Adisha Chopra. She's a senior manager, product line manager for Juniper Networks. Welcome. Thank you. Feels great to be here. So what do you think of this show? Have you been to a re-invent before? Oh my God, no, this is my first one. And I am so excited. It's the energy is so great. It's vibrant. I'm learning a lot. I'm very happy to be here. So Juniper's been around for a long time. Way predating this cloud, this whole cloud thing. So what are you guys up to? What's the latest and really why you're here at re-invent? What's your story at AWS? Yeah, yeah, absolutely. So I think the latest thing with us is, as early as today there was a, we were posted on the AWS partner solution website. Vodafone is partnering with Juniper for their SD-WAN offering with, you know, the SD-WAN controller and sitting in AWS, managing all their branch offices. So that's what's the newest with us. And, you know, we've been making waves with a lot of partnerships recently. Couple of months ago or maybe just a month ago we announced with Nutanix. So that, you know, that announcement was focused more for our enterprise customers. Integration with Nutanix is a hyper-converged infrastructure where Juniper will be, you know, an integral part of their networking, providing for their conversion infrastructure. And then before that, I think a few months ago we had Red Hat. We announced a partnership with Red Hat.  So as you were mentioning, Juniper's been around for a long time.  Now, telco cloud, right? And similarly for enterprise. If you think about it, you know, large enterprises and telcos are not that different, right? So that's where we were at. And that's the more kind of, we're following the evolution like our customers are, right? They used to be telco, now they're telco cloud. Juniper, I think the newest thing with Juniper, to be honest, in technology I spoke about partnerships, but it's our cloud first strategy. That's what we have in mind. We are evolving with our customers, helping them in their journey for cloud adoption, cloud migration, right? It's a couple of sentences to say that, oh, we're helping our customers with cloud migration. But we're, you know, there's so many steps in between. They're very complex. You need a lot of hand holding and we are right there for our customers. So what does that actually mean? When you are, you know, saying that you're helping your customers, are they, are you working with them to bring them multi-cloud solutions from AWS and Microsoft and Google or? Correct, exactly. Can you give me a scenario or a use case? Yeah, yeah, absolutely. So what, like I was saying traditionally, Juniper was, you know, a hardware focused company. So our existing customer base, they bought a lot of big heavy boxes from us. And of course on top of it, came a world-class routing and switching software component, right, and it was all bundled up and sold together. Now, with, you know, they're moving towards the cloud, towards AWS, towards GCP, towards Azure. We want to be able to provide to them and who better to provide this service to them. We understand they're on-prem network. We understand cloud networking. We understand the transport in between. So what we're doing is, for our customers, we manage their existing on-prem networks, which, you know, a lot of our customers, you know, they're huge and they have a significant amount of footprints, global footprint, right? So we understand that we were able to connect them to the AWS, to the GCP, to the Azure, right? And the value proposition for them is that if they wanted to do it themselves, they have to understand, you know, three different or five different clouds, right? You have IBM, you have digital ocean. There's a lot out there, right? And getting the OPEX or getting the talent to be able to understand all these things and do the migration, it's hard, right? This is a complex problem to solve. So what Juniper brings to the table is we abstract it out. So, for example, I wanted to... Well, I just want to say, you know, one of the things that you're talking about here, and this is a total switch, if I'm right, is are you becoming a managed service provider? We do have a managed service provider. Because it sounds like you're going to be putting a lot more money into that side of the business versus the straight up product side of the business. Yeah, yeah, that's where we are pivoting from, you know, we want to, our perception used to be that we are a hardware company. Now we are a cloud-first company. We are a software company. So we're definitely pivoting towards the software-based solutions, software-based offerings. It's like your iPhone, right? Or your phone. You buy the hardware, but you're really buying it for the iOS or for the applications that run on it. Networking is following a similar paradigm now, right? It's the hardware boxes. They're definitely our bread and butter still. But it's the software now that's enabling and giving it all the cool factor and the innovation that's happening. It's all in the software. Contrail, that's our story for multi-cloud. That's one of our product offerings. So what Contrail does is, and I think that's what I was kind of referring to earlier, it gives you that higher level of abstraction where you don't have to worry about, am I, is my workload running in AWS? Is my workload running in GCP? It doesn't matter, right? You, as an enterprise or as a telco, we want you to focus on powering your applications, powering your services. We don't want you to worry about your infrastructure. That's our job, right? We want to completely hide all the complexity away from you and just let you do what, generate revenue. So as an application developer, right? So I'm an application developer and I use Azure, for example, right? And that's kind of my platform and I'm doing some interesting stuff with some scripting or I'm building, just a general new website or something like that with a couple different things. So as a developer at that level, I don't even know about Contrail. Exactly, exactly. Exactly, but I don't think Contrail yet extends up to that layer where it can manage everything across multiple clouds.  And we want to keep it that way. Contrail is there, Contrail is at that level. Contrail is going to provide the plumbing. So you as a developer, today everything, all developers are moving towards containers, right? So for example, the Red Hat partnership that I brought up earlier, that's focused on their Red Hat OpenShift platforms, their PADS service, which is a container-based service. Contrail integrates with Kubernetes. We integrate with Mesos, we integrate with Docker. So as a developer, when you employ these tools to write your code using a CI CD platform, Contrail is sitting right under it, giving you that connectivity. So for example, when you're developing your application and you deploy it, you deploy part of it in Azure, you deploy part of it in AWS, right? And you don't care where it goes. You use one for like bursting or something like that. Exactly, yeah, yeah. The rest of it on-prem. Correct. You know, it's distributed, right? So who's going to plumb it and make sure that it's giving you the results that you need? That's where Contrail comes in. Gives you that plumbing between on-prem, between AWS. So how is that different from Kubernetes as a whole?  How does Contrail kind of come in and work with Kubernetes? So great question, by the way. You know your stuff. So Kubernetes is orchestration for your workloads, right? It's services.   Now what Contrail does is it provides the networking between those Kubernetes pods. So let's say you want to scale up your application. Okay, you had 10 pods, now you want to go to 20. Kubernetes makes that decision for you that you need the 20 pods. And then Contrail is sitting under it, giving you the networking for those 20 pods. So when those 20 pods spin up, Kubernetes folks Contrail and says, hey, 20 more and these need to talk to those 10 pods that were already there, right? So Contrail is open source, right? Correct. Why haven't you donated it yet to the CNCF? We are part of CNCF. I know that. But fundamentally, if you want that to be pulled as much as you do, it's already open source. You might as well kind of get on that thread with the Kubernetes folks and start talking to them about how you make it part of the core distribution that then goes into six different distros. But something along those lines versus don't start your own distro, right? Don't start your own distro. But look at how you can become integrated into that Kubernetes stream that they stream. Exactly. Yeah, that is definitely something that, like you're saying, it's something that we want to do. That's the direction that we want to go at. But I think the actual decision is maybe above my pay grade. So I don't want to make a commitment here. So, you know. You sure want to follow up on slightly different track when you talk about cloud first and you answer the question, which is when you say cloud first, is that kind of the way you're going to market with your customers? Or is that the way you guys are looking at Juniper in terms of transforming the company? And it sounds like you said it's more of the latter really starting to reformulate Juniper as a cloud first. So how is that transformation going inside the company? That's a pretty significant shift from selling boxes and maintenance agreements and shipping metals. Yeah, we are definitely modernizing from within, right? But a lot of it is driven by our customers. Like I was saying, you know, they are evolving. They want to connect to the cloud. And you know, we obviously want to help them do that. As part of that, we want to be microservices based, right? Because we want to be able to support containers. These are just things that, you know, we need to do. Juniper is a leader as far as, you know, innovation in networking is concerned. So it was never a question of if we want to do this or if we want to go down this path or not, right? It's when, right? And we are definitely working day in and day out to make that happen. So, you know, a lot of our offerings, like recently we came out with our containerized SRX solution. SRX is our full feature, full service, next generation firewall. And we have containerized it, right? I believe it's the first offering of its kind. Containerized, host based firewalls. So, you know, innovative stuff happening all the time. Like you said, you know, it's definitely a herculean task but we are up for it and we're doing it. And I'm just curious when the customer conversations, you know, the hybrid cloud, multi-cloud, public cloud conversation, right? There's a lot of conversations. How do you take your customers down the path? Where do you see them, you know, trying to navigate and it's going to be a pretty complex world for a CIO trying to figure out what they're supposed to buy and not buy it, how to pay attention. Can I hit all the boosts? Right, right. You're at AWS in three days, I don't think so. I know, yeah. These conversations to be honest have been going on for the past couple of years, right? A lot of our customers, the intent is there to move to the cloud. And you know, we are trying to help them with it. So we, you know, we design with them. We design their network. We design their topologies. We handhold them, telling them how to do this, right? They're existing networks that they have. The complexity comes in because everything, like think of a company, right, a large company, it then goes ahead and acquires 10 more and they all have their own networks. They all have their own environments. VMware, Red Hat, Nutanix. So different kinds of environments now all need to connect to the cloud. You don't want them to be siloed.  So when we talk to our customers, that's what we tell them that, you know, with a Juniper-based solution, we have so many of them that work together in a cohesive way to give you that end-to-end connectivity. Secure automated multi-cloud, that's our mantra, right? And as far as, you know, engineering is concerned, engineering simplicity. If you come down to Juniper, it's plastered all over the walls, right? Engineering simplicity. We were really driving that message internally so that, and a lot of the CICD stuff, right? The way we want our customers to use it is how we're using it, so that, you know, that improves our quality, that improves reliability, and all those things. So in terms of hand-holding our customers, we talk, you know, we're there on the table day one. We talk to them about their design. I see that a lot of our customers currently where they're at is they are trying to connect to the cloud. They all want to move towards the containerized services. They know that's the right thing to do. They're not quite there yet, right? The intent is definitely there. They're playing with it, but in terms of being in production, we're still, you know, a little bit off. Not too much, but we'll get there soon, right? So we talk to them, we talk about, you know, how they can make their applications cloud-ready. There's a couple of ways to do it. You lift and shift, or, you know, directly move to go cloud-native. We have all these discussions with them. What fits their bill, right? What is good for them? What is it that's going to work for them? And then, you know, of course the connectivity piece, right? But with it, security, reliability, and scale, right? A company like Juniper, obviously, you know, Innovator and Networking, we solve problems for at a different level, right? For our much larger customers. So we talk to them about scale. We talk to them about, you know, reliable security is huge, right? You have a workload that you spun up on-prem.  You're going to have to replicate it, say, in AWS. When you replicate it, you still want the same security that you had on-prem to apply to this workload, which is now going to be in AWS. How do you do that? It's easy, with contrary, right? Because it's intent-driven. You specify the intent, in fact, you specified the intent when you brought up the first workload, and it captured it, okay, I'm supposed to talk to, you know, say, I'm workload-red, and I can only talk to other red workloads, and I cannot talk to the blue workload, something like that, right? So you specify the intent, and then when that red workload now comes up in AWS, it already knows that I wasn't supposed to talk to the green workload, so that policy and all the intent moves with that workload. And this is all done through contrary, right? And the other thing, that single pane of glass, I'm sure you've heard about it a lot today, right? The single pane of glass, you specify it one time. Again, the abstraction away from all those, you know, five clouds that you're working with, you specify the red workload, the policy for the red workload one time, and then it doesn't matter where you bring it up. Contrail will automatically apply it everywhere, and, you know, it's good to go. That's great. Well, Disha, thanks for coming. I certainly got the energy to attack this big problem, so I hope the universe is fortunate to have you. Thanks for coming on and sharing this story. Great, thank you for having me. It was wonderful talking to you guys. All right, Disha, Lauren, I'm Jeff. You're watching theCUBE. We're at AWS ReInvent 2018. Come on down, we're in the main expo hall right by the center. Thanks for watching.


Everybody forced here ripping 22 23 prison basketball choice 10 box number 10 tiered I'm gonna sip water check out the results anytime a late in sports cards calm We'll go ahead and get started double chocolate chip, too Did I miss that one? I guess I did I guess I did miss it anyway. Yeah They all look pretty good Surprisingly this past week the vanilla one was really good the vanilla. It's like vanilla wafer kind But it was just like a sugar cookie with vanilla like buttercream is very good very Like not super fancy it is very good and the lemon one anything lemon. I'm pretty much in on any lemon dessert I will I will wreck Lemon desserts are like the best like fruit flavor desserts, I think all right silvers of a la depot for the heat and John Stockton Adam Red choice auto is a Larry Johnson for the Knicks. They are definitely closed on Sundays There's got to be another place close to you that They just they rotate their menu, so you just close on Sundays It's tough man stuff. I don't know why is that a nebula? I don't think it's nebula purple something. Yeah, I thought I've had a nebula If we do hang on all right blue yellow green Lillard blazers. I know one way to find out put those damn cards shingoon rockets purple Malik Monk Kings color match to 20 Nice card Kings 17 of 20 Blue number to 49 tea wolves Austin Rivers and red to 88 Jane McDaniels Timberwolves Yeah, lemon cello is delicious That's stuff that I don't get you in trouble though. Yeah, I wasn't too rich. It's just very good I don't like it's just a very very very solid cookie. That's silver Tatum Celtics and Wilkins for the Hawks Red choice rookie auto Hornets is Bryce McGowan's Lemon flavored pretty much anything is I'm kind of in on Blue yellow green Yanis bucks Wiggins Warriors Robi Spurs red to 88 Duarte Pacers and Purple Caruso Bulls to 25 a 20 the Caruso I guess we're not gonna recap this one. It's gonna it's pretty pretty fast break. It'll behind here Sorry golf we can't help it This is what we do every Sunday to be fair to be fair every Sunday We do this at some point we talk about we talk about the cookie menu silver Justin Holliday Hawks Hey, it's high as Jones Grizzlies Hey, red choice rookie auto jazz Walker Kessler There he is you guys were talking about it now. He's here see see I said like I said it's a fast break It's not fast break. It's choice There he comes cease and desist sir Nice Kessler really good hit there for the jazz blue yellow greens of Lonzo Bobby Portis with it with a real with an actual green here Steph Curry Warriors green choice six of eight That'll do Don't get Brian P big congrats green choice stuff. We've got red to 88 BAM out of bioheat and blue to 49 Julius Randall Knicks nice Nick's color match to 15 to 49 on the I'll call him dr. J. That's all I'll call him. That's a run. That's a run right there. We're in it We did it No, they feel the choice That's baseball, baby That's baseball, baby Silver badly be a wizards and rookie Bryce McGowan's Hornets Red choice auto Hallie God as Sam would call him for the Pacers Tyrese Halliburton. There he is a guy that Might get two more pieces of hardware soon with the finals trophy and the finals MVP award Yoke itch nugs Martin Hornets purple to 20 Bradley Beale wizards nice parallel there Hey, what's up, Hawk? We're getting rated by Hawk. Let's go buddy. What's up? What up red to 88 Isaac magic and blue Conley jazz the 49 All right, it's a good time to tell y'all what we got breakin coming up here Nine to go on flawless everybody nine to go So Mike's been a bit better than it man, everybody's here. It's a party now. It's a party Silver Julius Randall Knicks and rookie Caleb Houston magic Red choice auto for the Bucks is Sam Cassell Blue yellow green Simons Blazers of Kung-Woo Hawks Jared Jackson jr. Grizzlies ready to 88 Dame color match blazers solid card in blue to 49 Herbert Jones pelicans That's what I'm talking about chicken. That's what I'm talking about What's up Carl Sanders Chicken got him silver THT jazz Collins Hawks Red choice rookie Otto Nuggets Colin Gillespie Also making us look like fools How we do blue yellow green Diallo pistons Jay Chantate Rockets Terry Easton Rockets rookie. Yeah red the 88 heat Nicola Yovitch rookie and A blue debook Suns 18 to 49 This has been a solid break.  That sounds insane, dude Got sliver Wiggins for the thunder Jalen Smith Pacers That would never even occur to me to do that. That's what I'm saying red choice auto Pat Riley Lakers Blue yellow green Horford Celtics Keeta Kings Wesley rookie spurs with red to 88 Chris Weber Kings And Garrison Matthews Rockets Drive by he's the Bruins so much Not even a Bruins fan, dude, I gave up on hockey Silver Harden Sixers morally defeated Rookie Silver Blake Wesley Spurs Red choice no red choice rookie on a Terry Easton Rockets Yeah, we got the game on It's low scoring but it's decent good game. I saw we scored on an error Usually that happens the Red Sox we get scored on by call it by having an error Blue yellow green Scotty Barnes Raptors looking our Clippers Barclay Sixers. I haven't I'm not super dialed into it Got red choice to 88 Clarkson Jazz and blue rookie Ryan Rollins Warriors to 49 You can't feed a line when he's in the cage Must be trying to make his own saying and steal yours Three good shirts that say that or I'm at the National have you seen our the shirts that were Pretty fire. I don't want to spoil. I'm like and I'm gonna talk about it. It's pretty cool Silver Kendrick Nunn Lakers and dirt Novitsky Mavericks every time I do something like that I'm like, I won't spoil it rich will walk in the door like guys I'm like, well, I can't override him Red choice rookie Otto Wendell Moore junior T Wolves like when I was like, oh guys There's a special guest breaker on Thursday and everyone's like, oh, is it this person this person and then rich just comes to chat I'm breaking Thursday All right. Well the build up for nothing. Yeah, there's not really a There's no more of a Red Sox Yankees rivalry right now. That's like a fan's rivalry. Yeah, like yes I feel like the players don't have that not that it's not a rivalry But like the players nowadays, it's it's not really there It wasn't like late 2000s early Late 90s early 2000s. Yeah. Yeah when the Red Sox finally when the Red Sox finally got good We yellow green sexton jazz Cameron Payne Sons purple rookie to 20 Trevor Keele's nicks ready 88 Lonzo ball bowls and Blue to 49 Herbert Jones Pelicans Yeah, I feel like we're like the Yankees are just as much as the on-field rivals as like the rays are Somebody's somebody's a sign a clown like a ride to come out and stir it up. Yeah, that always works All right, you got silver or bam out of bio and KG Green thunder rookies and Cecil signatures to eight Jalen Williams Yeah, yeah Yankees ash I was definitely a bigger rivalry than Yankees Red Sox right now on the field Three of eight J. Will for the for the Thunder nice hit. Yeah, sure. We yellow green okoro calves box Kings purple Montres Harold Sixers to 20 Red to 88 TJ Warren nets and Blue to 49 on Yaka Okangu Hawks Yeah, the Astros win every time but as far as like it being Circle it on the calendar. That's it We'll just show the two greens that we had the lowest numbered we had a J. Will rookie out of for the Thunder and The staff curry for the Warriors. Thank you very much everybody. We're on to the contenders full case now


module 140 in production analysis we are going to study the production cost why we study the production cost because previously we have studied that the producer is producing with the utilization of the various factors of production efficiently if the total output that he is having it exhausts the available inputs and that was related to the technical efficiency but either it is having any insight that the producer is getting certain form of the profit or it is feasible for the producer to be having a capacity to retain his business so when we have to assess that point when we see that the producer is either going to profit or going to loss or you have such a situation that he is not getting profit or loss and he has to make a decision about his future so the inputs a producer is using because his prices are very difficult or rare that he should be stable in the market and if we look at the inflation these days we can say that the prices of inputs they are fluctuating daily so whenever the prices of the inputs they will be fluctuating then the respective production cost it will be changing and without considering the production cost we cannot assess the profit of our entrepreneurial so when we look at its economic efficiency then we have to look at the production cost in total and we have to look at its different components in short we say that the total cost is basically it is just the submission of all the factors of production giving the contributing share with their prices means if he is using 5 inputs then the amount of every input that is being used is multiplied by its market price and when we will add up these all incurred expenditure of on every input that will give rise to the total cost but now here we have the concept if we look at our inputs nature is different we can say that we have a lot of inputs that are fixed inputs and in the same way we say that there are certain inputs that are variable inputs so when we talk about fixed input or variable input then they can be related to their term of the decision here we can take short or long term time space but in short if we will say that when the producer is having the power of decision related to the utilization of input we can say that it is a variable and when that power is not in his hand because he has taken the decision then after that it becomes a fixed asset so this the narrow difference which is fixed or short but generally if we look at the assets on which once investment has been done which is if a farmer has taken the farm of his land then this is a fixed asset fixed cost and in the second form we call it sunk cost similarly in the industry list if he has put a jean factory or he has put a textile industry or he has put a spinning unit so now he cannot change the plant size of his machinery very quickly so this is his fixed asset but in the same way if other assets are like this if they are once invested then they also come in a fixed form but mostly all those inputs which we change in short duration in which most of the labor can also be done but if we say here many times the entrepreneur hires the labor for a year or in the long term so the permanent labor if we look at it sometimes we take it in fixed form but generally the inputs that we are going to purchase in the form of water we are going to purchase in the form of chemicals, the fertilizer and if we talk about the spinning unit that he is buying the thread or he is buying oil for the machinery or the labor cost payment so all of these will be its variable cost now coming to that if we look at it now so if we explain it in the form of a table so we can say that now in this example if the producer is having this fixed cost which we have said that it is $50 now if we want to tell you in one more concept that many times it happens that a fixed asset like a landlord has bought a tractor similarly if we say that an industrialist has installed a machinery or plant so keeping in view the discounted values if we look at it now it tells us that every machinery or building has a life so looking at that life if it is 10 years or 20 years then we can divide the total asset price by 10 or 20 years and in that we can then further, according to the discounted rate we can also put the incremental rate of some things that every year its price is being devalued and what is its estimate covering inflation so it is not possible that we say that this fixed cost if we look at it after one year that it has bought a 1 crore machinery but since when it has to deal with its accounts then the entrepreneur will not put that 1 crore amount in the same year rather he will share the premium or opportunity cost which if he says that if he has bought that 1 crore building then if its life is 20 years then he will distribute it in the accounts of 20 years and then further include it with its discounted rates now the variable cost here is definitely attached with those inputs which he is purchasing to complete the production function now if we look here the time output units are being produced he has added 50 units fixed cost and added 30 units variable cost and adding these two the input he purchased he gave 3 output units and in this way there was the total cost of 88 now if we look at it again then we go from 3 to 4 output units again the fixed cost is the same for it but now to produce another unit he had to use some incremental part of input system and if we look from here now he has used 50 units compared to 38 and the total cost of his 50 plus 50 that is equal to 100 but for one additional unit of output if we look here the difference between 38 and 50 so additionally he had 12 rupees or 12 dollars amount so if we look here for one unit of output the additional cost added was 12 dollars so in this form if we look here then from 3 units of output to 32 units of output we have a table in which we have this respective all the variable costs are added and we develop the total cost of fixed cost and for each additional unit if it is going to 10 or 22 then we have checked the marginal cost and now if we have to calculate then we can see that in the start this marginal cost was greater and then further at certain points it reduced and then after attaining a certain minimum level it again started to increase so if we look at this now this is higher than low and then again high so this will show us that the marginal cost curve will be higher than low and again higher so if we look at this one then we will see the face of U now if we come to column 6 then it gives now if we look at the average cost column 6 and column 7 if we come to column 8 then it gives the average total cost and if we look at the average total cost then that will be equal to this total cost means this was this 88 divided by this 3 so this column is this 8 this is equal to column 4 divided by column 1 and now where we see that this average total cost this total cost was 100 so it was divided by this 4 so average cost it was 25 so mostly when we are reading the average total cost in our different books then we will only have the name of this in the face of the average cost now if we say that what is the average fixed cost so we have these 2 columns fixed cost and variable cost so when fixed cost is divided by the output and when variable cost is divided by the number of the output it will be giving us the average fixed cost and average variable cost so if we look at all these scenarios then all types of the cost they basically depend upon the number of the outputs produced likewise the revenue when we calculate then price into amount of the revenue output produced so that is why economic profit or economic cost or economic efficiency is just depending on the amount of the output produced thank you


We're now going to work an example problem involving drag on an object. What we're going to consider is a bus that is moving along a surface at a given velocity. So we have a bus that is moving along at a velocity at time and the velocity is going to change. And so what we are given, so we're told that we have a bus traveling at v0. So it's moving at initial velocity v0 and then what we're told is that the engine fails and there is zero rolling resistance between the road and the bus. So all of a sudden the engine fails on this bus and it is going to decelerate and the reason why it's going to decelerate is because there is drag acting on the bus. So the information that we're given about the bus, we're told that the bus has mass m. So we know the mass of the bus is m and it turns out that it has a drag coefficient cd. We know the frontal area of the bus with which the drag coefficient is calculated. So that is given to us. We know the density of air. So it's going through some atmospheric air. I won't put it in infinity, I'll just put rho. So we know the density of the air and I think that is everything that is provided to us in this problem. So what we want to do is we want to be able to find an expression for the velocity. So we're looking for velocity of the bus as a function of time. So the way that we're going to do this is we're going to apply f equals m a to the bus. So let's draw out a free body diagram. Now we can neglect rolling resistance. So the only force acting on the bus is going to be the drag force. And the drag force we know is going to be the drag coefficient times the dynamic pressure times the frontal area. And this is what we're after in the problem, we're after the velocity.  And given the bus is moving in this direction, we're going to take that as being the positive x direction. So the force on the bus is going to be acting in the negative direction. And that would then agree with the bus decelerating through the velocity is going to get lower with time. So what we're going to do, let's plug in our drag force. So we end up with this differential equation here. And remember what we're after. We want to know what is velocity as a function of time. Boundary conditions, velocity at t equals zero, we were told was equal to v naught. So that is what we have to solve this problem. So let's rearrange a differential equation, get dv and v on the same side of the equation. And then let's work through it. So rearranging first of all, before we put dv and v together. Okay, so we've got v squared there. Now we can rearrange. And these are all constants. So I'm going to pull them outside of the integral sign. And on the right hand side, we're integrating. And the limits of integration, time, we're trying to find velocity as a function of time. And we were told the velocity at t equals zero. So we're going from zero to some arbitrary point in time. And the velocity is going from v naught to v, where v is the velocity at some point in time. Which is what we're trying to solve. So what we're going to do, let's integrate this equation. So we get this equation here. Now what we're going to do is just go and do some algebra, rearrange things. So let's walk through that. So we get this expression here. And remember what we're after is velocity as a function of time. So let's solve for that. And so we get this relationship here. So this is the velocity as a function of time of the bus after the engine fails. The only thing slowing it down is going to be the drag force because we're neglecting rolling resistance. And let's plot this just to get an idea as to what it looks like. Look at the extremes that we have for this equation. So if we have t equals zero, we can see that the second term in the denominator goes to zero. And so what we end up with is v of t equals v zero, which is the original velocity. That's what we should have. And another limit that we have, let's say time goes to infinity. As time goes to infinity, the second term in the denominator here is going to get very, very large. And consequently, velocity time equals infinity will equal zero. Asymptotically it's going to approach zero. So what we have are those as being two of the limits in our function. So I'm going to plot velocity and we'll plot time down here. We said this here was at t equals zero. So we begin here v naught and we know way out here we're eventually going to get to zero. And the other thing that we know just looking at this function, velocity of time is proportional to one over time. So it's an inverse relationship with respect to time. So not knowing exactly what it's going to look like, but I'll just sketch it in. Our velocity with time is going to look something like this. So that would be a curve then of our velocity as a function of time of the bus due to drag. Inverse relationship with time. So that gives you an example of a fairly simple problem dealing with drag. But it enables us then to calculate the forces and we can evaluate the velocity of this bus over time.