If you throw a ball straight up, there will be a point when it stops moving for an instant before coming back down. Consider this as we study the rate of change of human cannonballs in this lesson.
Rate of Change Review
X is changing as a function of time linearly, so the velocity is constant
Let's review. The rate of change is how one variable changes as a function of another variable. We can see this as how, for example, my location changes as a function of time. This is like a velocity.
Rate of Change in the Human Cannonball
So what happens in the case of the human cannonball? The human cannonball is launched at 35 mph. His rate of change at launch is 35 mph. When he reaches the apex, the top of his flight, his velocity is actually 0 mph. His rate of change is 0. When he starts plummeting back toward Earth, his rate of change is negative. His velocity is negative. So what happens if I zoom in really close to a point on the curve of his height as a function of time? If I look really closely, I can see that the slope at any given point on this curve is the tangent at that point. This tells us that the rate of change is not only equal to the velocity, it's also equal to the tangent to the curve.
The second set of curves shows velocity increasing over time
Examples of Changes in Velocity
Let's look at a couple of examples. If your position, x, is changing as a function of time linearly, so there's a straight line on this position-time graph, then your velocity is always constant. Say here that your position is changing as a function of time at about 35 mph. The slope on this x-t graph is 35. Your velocity is 35 mph.
Let's say that your position is changing like this as a function of time. When you start out, your slope, the tangent to this curve, is zero. But as time moves forward, the slope of the tangent to your graph is increasing. So your velocity is increasing over time. This is kind of like how if you start driving, initially, it's hard to get from 0 to 10 mph. But once you get to 10 mph, you can speed up really fast.
What if your position isn't changing as a function of time - you're just sitting still? Well, your velocity then is just zero, because the slope of your position as a function of time is zero. That makes sense. I'm just sitting there and position isn't changing as a function of time, so my velocity is zero.
In the final example, the slope of your position as a function of time is zero as is the velocity
Let's review. The rate of change is your velocity. It's how fast one variable is changing as a function of another variable. Usually, we think of this as how fast your position is changing as a function of time. It is also equal to the tangent to a position curve. If you graph your position as a function of time, the tangent anywhere on that curve is your rate of change, which is your velocity.