[MUSIC] Now, it's time to quantify acceleration and to relate it to velocity, time, and displacement. If you've started the experimental investigation, you'll probably have noticed that velocities and accelerations can vary in complicated ways. Is. Here's an example. Here's the train we saw last week. It's accelerating from rest in a straight line. It's velocity increases with time, but not in a simple way. Here's a simpler system in the lab. Once again, we have a slider on an air track, but this time there's a gentle slope. Again, we plot displacement versus time. Using the technique we used last week, we can plot velocity versus time as well. Remember, velocity is change in displacement over change in time, so it's the slope of the displacement graph. Slope equals rise over run. Here, it's a small slope, so a small velocity. Here, the velocity is large because the slider has accelerated. The initial slope gives us the initial velocity and, of course, the slope increases in time. The slider is accelerating. It's getting faster over time. We define acceleration as the rate of change of velocity, the slope of the velocity graph. And here, it looks as though the acceleration is pretty constant. The units of acceleration, velocity over time, are meters per second squared. Again it's a human scale unit. You can accelerate at a meter per second per second. Well, at least you can for a few seconds. And, as we will see next week, you can accelerate objects at nearly 10 meters per second per second, just by letting them fall. So, let's consider a case with constant acceleration. Constant acceleration, so the velocity increases at a constant rate. VFT is a straight line, starting at v0 and with slope equal to the acceleration, a. So, it has the equation v equal v0 plus at. That's another important equation and we'll want to remember it. We can interpret the a times t term. It's the area under the acceleration curve, a times t. Earlier today, we saw how to get from velocity to change in displacement. We take the area under the velocity time curve. Let's do that here. First, there's the rectangle with area v0 times t. Then, because of the acceleration, there's a right-angle triangle whose area is just half of width times height, 1/2 of at times t or 1/2at squared. So the displacement, x, has three terms x0, the initial displacement plus v0 times t, which is the area of the rectangle part under the v of t graph. And finally, 1/2at squared which is the area of the triangle part under the v of t graph. It's easy enough to interpret these terms. X0 is its initial position, and then there's an increase v0t, due to its initial velocity. We have that term even in the absence of acceleration. And finally, there's a term half at squared, because this object is accelerating. Let's collect the equations from these three plots showing constant acceleration. A equals constant. V equals v0 plus at. X equals x0 plus v0t plus 1/2 at squared. By the way, we don't use calculus in this course. But in calculus, the slopes of curves are obtained by differentiation and the area under curves by integration. If you'd like to see kinematics presented with calculus, we have a resource for that, and we have another resource which is an easy introduction to calculus. In real life, acceleration is almost never exactly constant. But we can often make the constant acceleration approximation. So you need to be able to write down these equations. But, you don't need to learn them like a parrot. Instead, think about each term as you write. Velocity starts at its initial value v0, but, if there's a positive acceleration a, v increases with time, so we add at. Displacement starts at initial value x0, but if there's a positive initial velocity v0, it increases with time, add v0t. Finally, if there is acceleration, then there's that, 1/2at squared term, where the last term is the area of a triangle with length, t and height, at. Now let's apply these same equations to this situation. The slider starts off with positive velocity in the x direction, but it slows down and then reverses. Traveling to the right, v is positive, but its getting smaller. It is decelerating, the slider has negative acceleration. Let's graph these equations. The x values starts at x0 and gets larger, but the rate of increase in x slows to 0 when x gets to its maximum. Then x decreases with time. To get the velocity, we look at the slope of the displacement time graph. Initially, v is large and positive, but it decreases to 0 and then becomes negative. Finally, we look at the slope of v of t. Here, it's negative and constant, so we have negative acceleration which in this case is constant. Incidentally, we went from displacement to velocity to acceleration by looking at the slopes. Previously, we went in the other direction by looking at the area end of the curves. We've spent some time on these equations because they're so useful as you will find out when you do the problems in this quiz.
