 Welcome to the screencast where we're going to look at an alternative form of the average velocity formula that will turn out to be very handy for us later. So we know that the average velocity of a moving object on a time interval that starts at t equals a and ends at t equals b is given by this formula. That's s of b minus s of a divided by b minus a, where s here is a function of time that tells the position of the object at time t. So the numerator is a difference in positions, which tells how far the object is traveled, and the denominator is a difference in time values. As we mentioned before, average velocity doesn't tell us the velocity of the object at any single point in time. Rather, it takes two time values and gives us an overall central value of the velocity over that time period. What we're going to do now is look at this formula from a different point of view, focusing on the ending time, b. In an average velocity problem, we look at the change in position starting at time a and ending at time b. Now another way to think about the ending time is that it's the starting time plus a small time change. For example, in the previous video, we looked at an average velocity problem over the interval from 0.2 to 0.5 seconds. We could just as easily have phrased this as looking at the time interval starting at 0.2 seconds that was 0.3 seconds long. In other words, we can specify a time interval by giving the starting and ending times, or we can give the starting time and specifying a small change in elapsed time. So let's keep calling the start time a, and let the change in elapsed time be denoted by the letter h. That makes b, the ending time, equal to a plus h. That's the start time plus the change. If we look at it from this point of view, we can update the formula. Here, I've taken the average velocity formula that we already know and just replaced b with a plus h in red. This happens in three places, in the subscript on the left hand side, in the numerator, where there used to be an s of b, and in the denominator. Now notice in the denominator, we can simplify a bit because the a and the minus a cancel each other out. That leaves us with this alternative formula here in terms not of two time values, but in terms of a starting time value and a change in time. This computes the same thing as the previous formula, but it doesn't from a different point of view, and it turns out in very short order it's going to be much easier for us to use. So to see how this version of the average velocity formula works, let's do a couple of examples. So let's see this tweaked version of the average velocity formula in action. We're going to do an example that's related to the examples from the last video. In that video we did an example of throwing a baseball straight up in the air from the top of a 10 foot platform, and we said in that case its position formula, s of t, is negative 16 t squared plus 20 t plus 10, where s is measured in feet, t is measured in seconds. Let's use the new average velocity formula to calculate the average velocity from time 0.2 to time 0.21. This is a very short time interval, and the length of that time interval is really the main player in this formula here. So I've just copied down the new version of the average velocity formula here. The average velocity on a time interval that starts at a and ends at a plus h is s of a plus h minus s of a divided by h. Now remember a is the starting time here, and so that's going to be 0.2 in this case, and h is the change in time, or the amount of elapsed time over the course of the interval. Now we start at 0.2, and we end at 0.21, and so the difference between those two is the elapsed time. That makes h equal to the difference between those two values. That's 0.01 seconds. Okay, so h gives the elapsed amount of time, and now I have enough information to put into my formula here. So the average velocity from 0.2 to 0.21 is going to be s of a plus h, which is 0.21 minus s of a, which is 0.2, and this is all divided by h. I've gone up to the site here and cooked up the numbers for the numerator here. You can do that and should do that on your own. Just use the formula up here at the top with the particular values. s of 0.21 is 13.4944, that's the decimal point there, and we already knew from a previous calculation where you could just redo it that s of 0.2 is 13.36, and this is all going to be divided by h, which I should have written in here. I wrote h, but I really should have put 0.01, so 0.01 here as well. Now let's just calculate the subtraction problem in the numerator, and that comes out to be 0.1344, and this is divided by 0.01, and so that gives us an average velocity on this time interval of 13.44 feet per second. So for our final example, we're going to look at something a little bit different here. We're going to keep the same physical setup where we're throwing the baseball up in the air, and we're going to use the same position formula as before, of course, but this time, instead of finding the average velocity on a time interval where both the starting and the ending times are specified, which is what we've done every single example we've seen, we're going to specify the starting time of this time interval to be 0.2, but we're not going to specify exactly when the ending time is. We're just going to find the average velocity from time equals 0.2, so we're going to nail down that starting time, but not the ending time, just to time equals 0.2 plus some small change in h. What we're going to end up with here is a formula for average velocity that is in terms of h only. This is going to be pretty useful for us in the next video. Let's see what we can do with this. The average velocity from 0.2 to 0.2 plus h, if I just use my formula, that would be s of the ending time, which is 0.2 plus 8. I'm going to subtract off s of 0.2 and then divide by the change in time, which is h in this case. Again, that's the ending time minus the starting time as my h value. There are a few things I can do. At least one step that I can do right off the bat here that doesn't require a lot of work other than copying down some stuff. I don't really know exactly what s of 0.2 plus h is right now, but I do know from previous experience that s of 0.2 was 13.36. Even if I didn't know that, I could just again go back up to my s formula and calculate it. I don't really know anything else here. What I'm going to do now is see if I can use the formula to expand and simplify this fraction that I have here. Let's go over to another slide where I've got this partially set up. I'm going to calculate s of 0.2 plus h. I'm just going to go into the formula for s and replace t with 0.2 plus h everywhere. So I'm going to have negative 16, 0.28 plus h squared plus 20 times 0.2 plus h plus 10. Now let's do some algebra to simplify this. The first thing I will do is foil out this quadratic term right here. I'll leave the checking of the details to you, but this comes up to be 0.04 plus 0.4 h plus h squared. Now distribute the 20 throughout the group here, and that will give me 4 plus 20 h plus a 10 at the end. Now I think the next algebra step would be to distribute the 16 throughout the group it's attached to. It's going to give me negative 0.64 decimal point there minus 6.4 h minus 16 h squared. There was also remaining a 4, a 20 h, and a 10. Now let's collect like terms here. I'll single underline the constant terms. There's one, there's one, and there's one. Those terms add up to be 13.36, which is interesting because that's what s of 0.2 was on the previous slide. Double underline those linear terms. There's one, and there's one. Those two terms add up to 13.6 h. And then finally there's only one quadratic term, 16 h squared. So in the average velocity form, if I could just page back there where I have the red bubble here at the bottom, that can now be replaced by this expression we just created. We'll do this final slide here and just try to put some pieces together. Now if I do the replacement that I just mentioned, I'm just going to copy down what we calculated. That's 13 point, let me do this in a different color so you see it a little bit better. It's like blue. 13.36 plus 13.6 h minus 16 h squared. And then we were subtracting off of 13.36. This is all being divided by h. Now the thing to notice here is that this 13.36 and this point 13.36 are opposites. So they're going to cancel off. And then I'm going to be left with the following. I'll have 13.6 h minus 16 h squared over h. Both of the terms in the numerator have a common factor of h. And so I'm going to factor off the common factor right what's left over 13.6 minus 16 h over h. And now I have two common factors being divided by each other and they will divide off. And so I have an interesting little simpler formula here for the average velocity. And that average velocity is just 13.6 minus 16 h. This is the average velocity from 0.2 to 0.2 plus h. So what this allows me to do is if I have any time interval at all that starts at 0.2, no matter how wide it is, if I know how long that interval is, that's the h value, I can very easily calculate the average velocity. And just as an example, let's redo the example we saw earlier. What is the average velocity from 0.2 to 0.21? Well in that case the h was equal to 0.01. And so the average velocity would just be 13.6 minus 16 times 0.01. That is 13.6 minus 0.16. And that gives you 13.44 feet per second, which is what we got through a considerably longer process earlier. But now if I wanted to recalculate the average velocity for some other value of h, it would be just as easy. And we're going to use that idea in the next and final video for this section. Thanks for watching.