Running from your little sister or just window-shopping, your speed is just a measure of how fast you move, or how your position is changing over time. In this lesson, learn about how velocity is a rate of change.
The Velocity of My Drive to Work
Graph for the morning commute example
Every morning I drive from home to work, but yesterday was a little bit different. On my way to work, I had a nice, easy pace and I had to stop at the stoplight in the middle. Overall, it wasn't too bad. When I got to work, however, I realized that I forgot my coffee at home. Now, I don't know about you, but when I forget my coffee at home, I'm a mess! So I turned around and drove home very quickly, and luckily, I didn't have to stop at the stoplight that time.
Let's take a look at what I did yesterday more mathematically. Let's call the distance I was from home x, and let's graph my position as a function of time. I started out at home, I got farther and farther away from my house, I had to stop at a stoplight and I was able to continue until I got to work. When I got to work, I remembered that my coffee was at home, and I drove straight home really quickly without stopping. If I take a look at the slope on this graph, the slope is the change in x divided by the change in t. This is the change in my distance divided by the change in time. The change in distance divided by the change in time is my velocity. This is also called the rate of change. Specifically, it's how fast my position (which is a dependent variable) is changing with time - in this case, my independent variable. But formally, we say that the rate of change is how fast one variable changes as a function of another variable.
Velocity and Inconstant Slopes
So where else do we use the rate of change? Let's consider a human cannonball. Let's graph the height of the human cannonball as a function of time. Here, the rate of change is how fast the human cannonball is rising up into the air as a function of time. You could also call this the upward velocity, or the vertical velocity, of the human cannonball.
Graph showing the velocity of the human cannonball
To recap, the rate of change is how one variable changes as the function of another variable. In the case of my driving to work, I was looking at how my position changed with time. For the human cannonball, we looked at how his height changed as a function of time.