Using Time as a Variable in Equations and Functions
In this video, we introduce the concept of using time as the independent variable in a function.
Using Time as a Variable
Up till now, we’ve almost exclusively used generic variables like x and y in our equations and functions. But now I want us to look at using time as a variable. If you look at physics, you have the equations of motion where position, velocity, and acceleration are functions of time.
If you look at biology or chemistry, you have exponential growth and exponential decay, which are both functions of time. Seeing time as a variable is something that will become more and more common as you move on in the sciences. Now I want us to look at a word problem that gives us some practice with creating functions where time is our variable.
Let’s look at a word problem example here. Let’s say we have two towns A and B and they’re separated by 300 miles. We have two trains that go between A and B. We have one train, we’ll call this train 1. It leaves at 2 p.m. and it travels sixty miles an hour. We have train 2 that leaves at 4:00 p.m. and it travels 120 miles an hour.
What we want to know is which train is going to arrive at town B first. The first step in doing this is going to be to set up our equations for the position of each train with respect to time. The first train leaves at 2 p.m., so 2 p.m. is going to be our t equals zero time. Time, in terms of our functions is going to start at 2 p.m.
The equation, or the function, for the position of train 1 we’ll call it P1 with respect to time. P1 with respect to time is equal to 60 miles an hour. 60 times t, where t is measured in hours starting at 2 p.m. This is our position equation for train 1. Our position equation for train 2, P2, with respect to time, is going to be 120, because train 2 goes 120 miles an hour.
120 times- it starts at 4:00 p.m. We have to give train 1 a head start. We have to write this as 120 times t minus 2, because train 2 doesn’t start until two hours after train 1. These are our two equations for the position of the two trains. If we’re looking for which train arrives at B first, there are a couple of different ways we can go about that.
One way we can go about it is to set both of these position equations to 300. That will tell us the time when both of these trains reach B. The time that is the earliest as the one that arrived first. Another way we could solve it is to set the two equations equal to one another, which would tell us when the two trains are in the same location.
If that location is before 300, that would mean that train 2 arrives first, and if it’s after 300, that would mean the train 1 arrived first. We’ll go ahead and do the first method where we set both equations equal to 300 and we’ll see which one arrived first. If we set the first equation equal to 300 it gives us 60 t is equal to 300.
T is equal to 300 divided by 60 or five hours. Five hours after 2 p.m. is equal to 7 p.m. Train 1 arrives at 7:00 p.m. Train 2, meanwhile, having left two hours later, also has to travel 300 miles. Let’s go ahead and multiply this out. To simplify, that give us a 120 t minus 2 times 120, or 240.
That’s equal to 300, or 120 t is equal to 540 or t equals 540 divided by 120. Let’s go ahead and simplify. We can divide both sides- or divide top and bottom by 10. We can divide top and bottom by two. That gives us 27/6. Then, we can further divide by three. That gives us 9/2 or 4.5 hours. 4.5 hours after 2 p.m. is 6:30 p.m.
That tells us that train 2 arrived half an hour before train 1. This is just one example of using time as a variable to solve a word problem. As I said, if you get much farther in science you’re going to encounter a lot of equations where time is your variable.