 Hello, and welcome to the screencast on calculating instantaneous velocity. Let's start with a review of the basic concepts involved. First, remember that the average velocity of a moving object can only be calculated over a time interval, and we need two time values to compute it. Here are the two formulas we've seen that compute average velocity. Namely, we need a start time and either a stop time or a time length to calculate average velocity. Also recall that an object's average velocity does not give us any information about how fast the object is moving at a single point in time. It only gives us a general central value of its velocity. For example, when Alice was taking five minutes to travel between two classroom buildings that were 100 meters apart, her average velocity on that time interval was 100 divided by 5 or 20 meters per minute. But that doesn't tell us how fast she was traveling at any single moment. She could have been stopped for a while and then ran, or she could have walked 20 meters per minute constantly throughout the whole time. The average velocity figure of 20 meters per minute does not tell us anything about that. When we calculate average velocity, we calculate it over an interval, and we need two different time values to feed into the formulas. By contrast, the instantaneous velocity of an object is how fast it's traveling at a single instant at a single moment in time. So when we calculate an instantaneous velocity, we speak of finding the instantaneous velocity at a certain time value, just one, not two of them. Now the idea behind computing instantaneous velocities is which we're going to begin to touch on in this video. The basic idea is to calculate the instantaneous velocity of an object at the single moment time equals a. We're going to find average velocities of that object over shorter and shorter time intervals that start or end at t equals a. And if those average velocities converge onto a single value, then that value is the instantaneous velocity. Let's take a look at an example. So let's look at this example where we are going to take an object and throw it straight up in the air, and its height above the ground is given by this formula s of t. It's equal to negative 4.9 t squared plus 100 t plus 5, and that is measured in meters where t is measured in seconds. And what we're going to do here is find the instantaneous velocity of this object that I'm throwing up in the air exactly at the time t equals 15 seconds. Now again, this is not an average velocity. This is an instantaneous velocity at a single point in time. So here's the process for how we're going to do this. We're first of all going to get a simple formula for computing average velocities from time equals 15 to time equals 15 plus h, where h is some unspecified length of a time interval. This is going to be very similar to the very last example from the last video that we saw. And then we're going to use that simple average velocity formula to look at a sequence of average velocities over intervals that start at time equals 15 and last for a shorter and shorter amount of time. The philosophy here is that the shorter the time interval, the closer to instantaneous the average velocity should get. As we're going to look at a sequence of average velocities for smaller and smaller values of h, both positive values of h and negative values of h. And then part three, we're just simply going to observe what those average velocities do. If they converge towards a single value, then we're going to call that the instantaneous velocity. So first things first, let's start with number one of this process of getting a simple formula for the average velocity from time equals 15 to time equals 15 plus h. So let's set this up. We're going to use the alternative formula that we looked at in the last video where we want to compute s of 15 plus h minus s of 15, all divided by h. Now this involves finding two pieces here. First go off to the side and calculate s of 15 plus h and then s of 15 and then just import them back in. Actually I think I'm going to do s of 15 first because that's fairly easy. s of 15 is just plugging 15 into the formula for s that we saw on the previous slide. I will let you do that and you come up with 402.5 as a result. s of 15 plus h is considerably longer so I'm going to use most of this slide to calculate it. Let's do that down here and change it back to black. So what is s of 15 plus h? Well I'm going to use the formula here so this is negative 4.9 times 15 plus h squared plus 100 times 15 plus h plus 5. And now I'm just going to do a number of algebraic steps here to simplify this down as much as possible. Let me start by using the foil method to expand out this quadratic term right here. I have a negative 4.9 and then 15 plus h, the quantity squared is 225 plus 30 h plus h squared. While I'm at it let me distribute the 100 throughout the group here and that of course is going to give me 1500 plus 100 h and then there's a 5 that's hanging on to the end here. Next, not quite finally but next I'll distribute this negative 4.9 throughout this group to all three of those terms. That will give me negative 1102.5, calculate that out, minus 147 h minus 4.9 h squared. And then I have the remainder from previous step which is 1500 plus 100 h plus 5. All the expanding and so forth has been done, now let me just find the like terms. Here's a constant term, there's a constant term, there's a constant term. Those all add up to 402.5 which again it's kind of nice because that's what s of 15 was. The linear terms here, the ones with a single factor of h there and there, those are going to add up to negative 47 h. And then finally I have one quadratic term here of 4.9 h squared. So on the next slide what we're going to do is just pull this result back into here and then try to simplify the fraction a little bit. So here we are with the average velocity formula again. Let me use a different colored blue for this. The s of 15 plus h we just calculated out to be 402.5 minus 47 h minus 4.9 h squared. That was this expression here. Now in green I'll put the s of 15, that s of 15 was 402.5. And now this is all divided by h. Again the green and the blue are from separate calculations. It's often helpful when you have a complicated formula that's in pieces to go off to the side and do each piece separately. Now the thing you notice here is that the 402.5 will subtract off from each other. And I am left with negative 47 h minus 4.9 h squared all divided by h. And the nice thing about that expression is that every term in the numerator has a common factor of h. So again I'm going to factor off the h and then write down what I have left over, negative 47 minus 4.9 h. This is all divided by h. And now since I have a factor of h on top and on bottom they divide off. And so my average velocity from 15 to any length time interval I wish is going to be negative 47 minus 4.9 h. And now we've completed stage one of this process of finding instantaneous velocity. We found a simple formula for the average velocities. So let's now use the formula for average velocity that we have that's very simple. And use it to help us calculate the instantaneous velocity of the object at time equals 15. I set up a table here below where I'm going to calculate five average velocities over intervals that start at time equals 15 and end at time equals 16, time equals 15.5, time equals 15.1, and so on. We're going to let the time intervals get smaller and smaller and see what the average velocities do. Well to use my formula for average velocities it depends on h. Now h is the length of the time interval. That is more precisely the ending time minus the starting time. The starting time is locked in at 15. The ending times are changing here. So let's do the first row of this table first. If the ending time is 16 and the start time is 5, that means that h is equal to 1. And so the average velocity, I'm just going to use my formula up above. That would be negative 47 minus 4.9 times 1. And that is equal to negative 51.9. These are all average velocities and so this is going to be measured in meters per second. Now let's move that one average velocity that does not tell us anything about the instantaneous velocity at 15. Remember, average velocities reveal no information at all about the velocity of the object at any single point in time, including the endpoint. So I can't just conclude here that the instantaneous velocity at time equals 15 is equal to anything. I don't have enough information. So let me proceed through my table and see if I can gather some information. In the second row I'm using an ending time of 15.5. So this is going to be an average velocity over a one half second time interval. Very short, but still got some length to it. Since it starts at 15 and ends at 15.5, the h this time is 0.5. And so my average velocity formula up above would tell me that the average velocity on that interval is negative 47 minus 4.9 times 0.5. This is using the h value and that comes out to be negative 49.45. Also in meters per second, although to save space, I will not write the units on these anymore. Now we can go through the rest of these table values and just observe what happens. Still, I'm not totally sure exactly what the instantaneous velocity is at time equals 15. I think we need to see a sequence of values and see what they all do. Let's fill in the h values for the remaining three. If the ending time is 15.1, then the h value is 0.1. If the ending time is 15.01, the h value is 0.01. And likewise the last h value is 0.001. Now you can go through and put those into your average velocity formula of above just as I've done for one and 0.5. I'm just going to skip to the results here and you can verify these independently. The average velocity on this time interval is negative 47.49. On the fourth time interval, where it's only a hundredth of a second long, the average velocity is negative 47.049. Finally, over the one-thousandth of the second long interval is negative 47.0049. Now, if you look at these values of average velocities, as you get shorter and shorter in your time intervals, it appears that these values are approaching something, possibly negative 47. There's one little detail we need to check here though before we make any conclusions about the instantaneous velocity. And that is what happens if I use a time interval that starts just before time equals 15 and ends at time equals 15. Well, that's what this slide is for. We're using the same average velocity formula, but now we're looking at time intervals that start before time equals 15 and get shorter. Now, I'd say start before time equals 15, but I have, it says end time here, and I'm doing that to stay consistent with the way the formula works. If I'm starting my time interval at time equals 14, and it's ending at time equals 15, that is technically an H value that's negative. It's like starting at 15 and going backwards in time to time equals 14. So the H value that I would be using for this row here would be negative 1. Likewise, this would be an H value of negative 0.5 because it's like starting at time equals 15 and going backwards in time 1 half of a second to 14.5. Now, let me fill in the average velocity values here that we can calculate. Just use the average velocity formula above. This would be negative 47 minus 4.9 times a negative 1, and that comes out to be negative 42.1, and again, this is in meters per second. With that H value of negative 0.5, you get an average velocity of negative 44.55, and you can do the work on that yourself. Just use the average velocity formula. Here, the H value is negative 0.1. This would be an H value of 0.01, and likewise, this would be 0.001, and the average velocity values that you get with these particular values of H, negative 46.51, negative 46.951, and then finally negative 46 plus 9951, 0.9951. Now, here's step three of the process. We're just simply going to observe what all of these average velocity values are doing as the time interval shrinks to zero length. In the first slide, we looked at the average velocity values seem to be pretty clearly approaching negative 47 meters per second, and that was also true of these values if you start at 15 and move backwards. These are also approaching negative 47 meters per second. Since the average velocities are approaching negative 47 meters per second, no matter which way that you approach time equals 15, we're going to say that this value here is the instantaneous velocity of this object at time equals 15. Thanks for watching.