 Hello, and welcome to this video on how to use the average velocity formula. Calculus is all about rates of change, and one of the simplest rates of change you know about is velocity, which refers to the rate at which a moving object is actually moving. Therefore, this is where we will begin our study of calculus, specifically by understanding the concept of average velocity. Conceptually, average velocity is very simple, and we use this idea every day. As a motivating example, suppose you have two classroom buildings on a campus that are 100 meters apart and are connected by a straight sidewalk. Alice leaves one of the classroom buildings that's precisely 2 p.m., and she arrives at the other building at precisely 2.05 p.m. How fast was she moving during her trip? Well, she covered 100 meters in 5 minutes, so this is about 100 divided by 5 or 20 meters every minute. That figure, 20 meters per minute, is her average velocity over this time interval from 2 o'clock to 2.05. This is called an average velocity because it gives an overall central value of Alice's velocity on the interval. Notice that it does not tell Alice's velocity at any single point in time during this 5-minute period. Maybe she met a friend coming out of the first building and stopped the talk for 3 minutes, then took off at a fast pace for the remaining 2 minutes. Or maybe she walked at a constant 20 meters per minute without speeding up or slowing down for the entire 5 minutes. We don't know. But her average over this time interval was 20 meters per minute regardless of what she actually did during that interval. With that basic concept in mind, let's look at the formula for average velocity and see how to use it. To calculate the average velocity of an object, we have to have not a single point in time, but two points in time, a start value and a stop value. Let's call the start time A and the stopping time B. We also have to have a way of knowing where the object is at the start and stop time values. For now, let's assume that we have a function called S of t that gives the object's position S at any time t. With that information, the average velocity of an object from time t equals A to time t equals B is given by this formula. Let's examine the formula for a minute and see how it works. On the left side, the letters A, V just stand for average velocity. And notice there's a subscript below it that gives the interval A to B. This denotes the interval of time starting at time A and stopping at time B. On the right side of the equation, we have a fraction. In the denominator of that fraction, we have B minus A. That is a difference of two time values, and what it tells us is the length of the time interval measured in time units like seconds or minutes. In the numerator of the fraction, we have S of B minus S of A. This is the difference in the positions of the object, the position of the object at the end of the time interval at time B minus the position of the object at the beginning of the time interval at time A. So now we've seen the structure of this formula. Let's look at three brief examples of how to use it. Let's begin with our Alice example from earlier in the video. We can arbitrarily set her starting position outside the first classroom building equal to position zero. This would mean that her ending position at the other building is 100. To get her average velocity from two o'clock to two o'five, we'll take the ending position at time equals two o'five, that's 100, minus the starting position of zero, that would give us obviously 100. Then we'll divide by the ending time minus the starting time. That difference is five minutes. So her average velocity from two o'clock to two o'five is 20 meters per minute just as before. Notice that the units of average velocity follow from the formula. Average velocity is computed as a fraction. The units in the numerator are distance units, in this case, meters. The units in the denominator are time units, in this case, minutes. So the units of the fraction are meters over minutes, which is better said, meters per minute. Now let's look at two examples using a more complicated formula. Suppose you have a baseball and you throw it straight up in the air from the top of a 10 foot tall platform. Its position in feet above the ground at time t seconds is given by the function s of t equals negative 16 t squared plus 20 t plus 10. What's the ball's average velocity? Over the time interval, starting at t equals 0.2 seconds and ending at t equals 0.5 seconds. Let's set up the formula first. So a v of 0,2 to 0,5, the interval, is the fraction, s of 0.5 minus s of 0.2. Again, this is the position at the end of the interval minus the position at the beginning of the interval, divided by 0.5 minus 0.2, which is the ending time minus the beginning time. In the denominator, the arithmetic is easy, 0.5 minus 0.2 is 3. In the numerator, we go to the formula for s to work out these two positions that we need. S of 0.5 is negative 16 times 0.5 squared plus 20 times 0.5 plus 10. Which I'll let you work that out, but it equals 16. And likewise, s of 0.2 works out to 13.36. So in the numerator, we have 16 minus 13.36, which is 2.64. That makes the average velocity 2.64 divided by 0.3, which is equal to 8.8. The units on this number are given by the fraction again. Since the numerator is in feet and the denominator is in seconds, the average velocity is 8.8 feet per second. This final example will show you that the order of subtraction matters in the average velocity formula. Let's take the same baseball example as we just did and find the average velocity from time equals 0.5 to time equals 1. This time, I want you to work out the average velocity yourself first. Pause the video, work out the average velocity from 0.5 to 1, and then write it down. When you're done, unpause the video for the answer and a debrief. So hopefully, you got negative 4 feet per second as the answer. Here's why, and then let's discuss the negative sign and what that might mean. So the average velocity on the interval 0.5 to 1 is s of 1 minus s of 0.5 divided by 1 minus 0.5. On the bottom, of course, of this fraction, we're going to get 0.5. On the top, we need to compute s of 1, and that is negative 16 times 1 squared plus 20 times 1 plus 10, which works out to 14. And earlier, we calculated s of 0.5 already, and we know that that's equal to 16. So on the top of this fraction, we have 14 minus 16, which is a negative 2. And negative 2 divided by 0.5 is negative 4. So we get the negative sign on this velocity because on the average, the ball is moving in a downward direction rather than an upward direction on this time interval. Time equals 0.5 at the beginning of the interval. The ball starts at 16 feet in the air. But it ends lower than this at 14 feet at time equals 1. So over this interval, it lost two feet of position over the interval. And so the negative change of position leads to a negative velocity. So a takeaway from this example is that velocity actually has a sign value. A positive velocity means forward, generally speaking. And a negative velocity means backward, generally speaking. And the direction of travel really does matter. That's it for this video. Thanks for watching.