Calculus: The Velocity Problem

let's start our investigation of calculus by looking at something that I call the velocity problem so here's the idea imagine I'm driving down the highway and I say that at 2:00 o'clock I'm at the hundred mile marker but that at 2:15 I'm going to be at the hundred and ten mile marker so the question is how fast am I going but the real question is what do I mean by how fast am I going am I asking how fast am I going at two o'clock am I asking how fast am I going at 2:15 am i asking how fast I'm going over this interval we have to be more precise about what it is that we're asking one way that we could answer it is by something called the average velocity so the average velocity is going to be the change in the distance divided out by the change in the time or if I want to use a little bit of mathematical notation I could say that this is Delta V for the change in the distance divided by delta T the change in the time so let's compute this average velocity over this 15-minute time interval that we have so if I'm going to come down and try to figure out what the Delta D is going to be well we went from the hundred to the 110 miles so 110 minus 100 and the unit's here are going to be miles and then if I want to investigate the change in time well I've gone from 2 o'clock to 2:15 so language right they said to 15 minus 2 o'clock I'm sort of abusing my notation here a little bit but I can do this computation to say that the difference here is going to be 10 miles divided out by 50 immense now this answer is actually perfectly acceptable it's just that we don't typically give speeds in in miles per minute we normally give them in miles per hour so I can do a little unit conversion here and I can say are going to be sixty minutes inside of one hour and then the minutes are going to cancel and this is going to be 600 divided by 15 and so this is going to be 40 miles per hour so this is one way to answer this this is the average velocity over this 15-minute time interval now I'm going to ask a slightly different question its what would a cop measure at exactly 250 we had previously decided at the average velocity over this 15 minutes was 40 miles per hour so how do we answer the question what is the cop going to tell us that velocity is at 215 exactly well we don't know how to answer this question with the amount of information that we have for example suppose I was going way faster I was speeding but I was speeding only from say 2 o'clock until 2:10 and then we get a traffic light and we slow down the cop might not measure us having this higher speed because the average velocity was going to be 40 miles per hour even though at some point in the middle we've been traveling way faster than 40 miles per hour and at some points we're traveling way slower than 40 miles per hour so this problem it turns out that we have not enough information in terms of that's essentially two different concepts here the one that we had seen before was called the average velocity and it was over an interval but what we're really asking here when I say what is the cop measuring at 2:15 precisely something called the instantaneous velocity and this the average velocity over an interval in the instantaneous velocity at one exact time are indeed different things I'm going to give you a little bit of a different way to think about the instantaneous velocity suppose they had a whole chart of information here and what I want you to note is that what's going on here in this time interval column is it I'm getting smaller and smaller and smaller time intervals so the first rule here is the two to the two fifteen but but then I do to ten to two 15 to 14 two to 15 I go all the way down until I'm like only a one-second interval to 14 and 59 seconds all the way up to to 15 so then for each of these time intervals we know if we have a time interval we can compute the average velocity so I'm just going to imagine we've gone and done that and we've got 480 46 going down and then in this last second here this one tiny little time interval it appears that we're going 60 miles per hour over that time interval now if you had if you were a cop and you had this table of data of our different average velocities but do you think that you should be getting a ticket at the 60 miles per hour if you're driving in the city set well if you only look down here at the bottom at that last second and you see that this average velocity is 260 miles per hour you can be really confident that at the time of 215 that you're going to see some speed which is going to be at least faster than whatever the speed limit is something very very very close but not exactly but very close to 60 miles per hour so this leads me to a notion of a sort of limiting process if I want to go the instantaneous velocity that is the velocity exactly at 215 then what I'm going to do is look at smaller and smaller and smaller time intervals where the time interval is getting really really really close to 215 and then you can imagine we could carry on if we had more accurate measurements this table could carry on going perhaps forever as our intervals got closer and closer and closer to 215 and then our average velocities would presumably get closer and closer to some number so we can think of instantaneous velocity the velocity exactly at 215 to be analogous to the sort of limiting processes these average velocities are done over smaller and small and smaller intervals indeed this is effectively how modern lidar guns are going to work that police have when they're trying to measure what your speed is they don't tell you the exact velocity a specific time exactly they interpret it as a very close approximation the way it works is this that they can send out this pulse of light and they can figure out what the distance between the cop and the car is going to be and then some tiny fraction of a second later another pulse is sent out and they can again get a new reading of where exactly this distance between the cop and the car is going to be so what you have is a tiny tiny tiny tiny little interval because the time between these pulses is really small and then you get a tiny change in distance as well because there hasn't been much changed the car hasn't moved a lot but nonetheless you get two distances and two times and they can compute the average velocity over that interval and in Kreutz they send out a whole number of these different policies really really really really quickly and then you can do some sort of average to try to get a very accurate reading so this computing of average velocities is a very important process for us in the limit as our time intervals get small

let&#39;s start our investigation of calculus by looking at something that I call the velocity problem so here&#39;s the idea imagine I&#39;m driving down the highway and I say that at 2:00 o&#39;clock I&#39;m at the hundred mile marker but that at 2:15 I&#39;m going to be at the hundred and ten mile marker so the question is how fast am I going but the real question is what do I mean by how fast am I going am I asking how fast am I going at two o&#39;clock am I asking how fast am I going at 2:15 am i asking how fast I&#39;m going over this interval we have to be more precise about what it is that we&#39;re asking one way that we could answer it is by something called the average velocity so the average velocity is going to be the change in the distance divided out by the change in the time or if I want to use a little bit of mathematical notation I could say that this is Delta V for the change in the distance divided by delta T the change in the time so let&#39;s compute this average velocity over this 15-minute time interval that we have so if I&#39;m going to come down and try to figure out what the Delta D is going to be well we went from the hundred to the 110 miles so 110 minus 100 and the unit&#39;s here are going to be miles and then if I want to investigate the change in time well I&#39;ve gone from 2 o&#39;clock to 2:15 so language right they said to 15 minus 2 o&#39;clock I&#39;m sort of abusing my notation here a little bit but I can do this computation to say that the difference here is going to be 10 miles divided out by 50 immense now this answer is actually perfectly acceptable it&#39;s just that we don&#39;t typically give speeds in in miles per minute we normally give them in miles per hour so I can do a little unit conversion here and I can say are going to be sixty minutes inside of one hour and then the minutes are going to cancel and this is going to be 600 divided by 15 and so this is going to be 40 miles per hour so this is one way to answer this this is the average velocity over this 15-minute time interval now I&#39;m going to ask a slightly different question its what would a cop measure at exactly 250 we had previously decided at the average velocity over this 15 minutes was 40 miles per hour so how do we answer the question what is the cop going to tell us that velocity is at 215 exactly well we don&#39;t know how to answer this question with the amount of information that we have for example suppose I was going way faster I was speeding but I was speeding only from say 2 o&#39;clock until 2:10 and then we get a traffic light and we slow down the cop might not measure us having this higher speed because the average velocity was going to be 40 miles per hour even though at some point in the middle we&#39;ve been traveling way faster than 40 miles per hour and at some points we&#39;re traveling way slower than 40 miles per hour so this problem it turns out that we have not enough information in terms of that&#39;s essentially two different concepts here the one that we had seen before was called the average velocity and it was over an interval but what we&#39;re really asking here when I say what is the cop measuring at 2:15 precisely something called the instantaneous velocity and this the average velocity over an interval in the instantaneous velocity at one exact time are indeed different things I&#39;m going to give you a little bit of a different way to think about the instantaneous velocity suppose they had a whole chart of information here and what I want you to note is that what&#39;s going on here in this time interval column is it I&#39;m getting smaller and smaller and smaller time intervals so the first rule here is the two to the two fifteen but but then I do to ten to two 15 to 14 two to 15 I go all the way down until I&#39;m like only a one-second interval to 14 and 59 seconds all the way up to to 15 so then for each of these time intervals we know if we have a time interval we can compute the average velocity so I&#39;m just going to imagine we&#39;ve gone and done that and we&#39;ve got 480 46 going down and then in this last second here this one tiny little time interval it appears that we&#39;re going 60 miles per hour over that time interval now if you had if you were a cop and you had this table of data of our different average velocities but do you think that you should be getting a ticket at the 60 miles per hour if you&#39;re driving in the city set well if you only look down here at the bottom at that last second and you see that this average velocity is 260 miles per hour you can be really confident that at the time of 215 that you&#39;re going to see some speed which is going to be at least faster than whatever the speed limit is something very very very close but not exactly but very close to 60 miles per hour so this leads me to a notion of a sort of limiting process if I want to go the instantaneous velocity that is the velocity exactly at 215 then what I&#39;m going to do is look at smaller and smaller and smaller time intervals where the time interval is getting really really really close to 215 and then you can imagine we could carry on if we had more accurate measurements this table could carry on going perhaps forever as our intervals got closer and closer and closer to 215 and then our average velocities would presumably get closer and closer to some number so we can think of instantaneous velocity the velocity exactly at 215 to be analogous to the sort of limiting processes these average velocities are done over smaller and small and smaller intervals indeed this is effectively how modern lidar guns are going to work that police have when they&#39;re trying to measure what your speed is they don&#39;t tell you the exact velocity a specific time exactly they interpret it as a very close approximation the way it works is this that they can send out this pulse of light and they can figure out what the distance between the cop and the car is going to be and then some tiny fraction of a second later another pulse is sent out and they can again get a new reading of where exactly this distance between the cop and the car is going to be so what you have is a tiny tiny tiny tiny little interval because the time between these pulses is really small and then you get a tiny change in distance as well because there hasn&#39;t been much changed the car hasn&#39;t moved a lot but nonetheless you get two distances and two times and they can compute the average velocity over that interval and in Kreutz they send out a whole number of these different policies really really really really quickly and then you can do some sort of average to try to get a very accurate reading so this computing of average velocities is a very important process for us in the limit as our time intervals get small

Transcript for:Calculus: The Velocity Problem

Transcript for:
Calculus: The Velocity Problem