# Motion in One Dimension

## Velocity vs. Time Graphs


### Section Key Terms


### Graphing Velocity as a Function of Time

Earlier, we examined graphs of position versus time. Now, we are going to build on that information as we look at graphs of velocity vs. time. Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity; we will discuss acceleration more in another chapter. These concepts are all very interrelated.
What can we learn about motion by looking at velocity vs. time graphs? Let’s return to our drive to school, and look at a graph of position versus time as shown in .
We assumed for our original calculation that your parent drove with a constant velocity to and from school. We now know that the car could not have gone from rest to a constant velocity without speeding up. So the actual graph would be curved on either end, but let’s make the same approximation as we did then, anyway.
Looking at this graph, and given what we learned, we can see that there are two distinct periods to the car’s motion—the way to school and the way back. The average velocity for the drive to school is 0.5 km/minute. We can see that the average velocity for the drive back is –0.5 km/minute. If we plot the data showing velocity versus time, we get another graph ():
We can learn a few things. First, we can derive a v versus t graph from a d versus t graph. Second, if we have a straight-line position–time graph that is positively or negatively sloped, it will yield a horizontal velocity graph. There are a few other interesting things to note. Just as we could use a position vs. time graph to determine velocity, we can use a velocity vs. time graph to determine position. We know that v = d/t. If we use a little algebra to re-arrange the equation, we see that d = v    t. In , we have velocity on the y-axis and time along the x-axis. Let’s take just the first half of the motion. We get 0.5 km/minute    10 minutes. The units for minutes cancel each other, and we get 5 km, which is the displacement for the trip to school. If we calculate the same for the return trip, we get –5 km. If we add them together, we see that the net displacement for the whole trip is 0 km, which it should be because we started and ended at the same place.
The area under a velocity curve represents the displacement. The velocity curve also tells us whether the car is speeding up. In our earlier example, we stated that the velocity was constant. So, the car is not speeding up. Graphically, you can see that the slope of these two lines is 0. This slope tells us that the car is not speeding up, or accelerating. We will do more with this information in a later chapter. For now, just remember that the area under the graph and the slope are the two important parts of the graph. Just like we could define a linear equation for the motion in a position vs. time graph, we can also define one for a velocity vs. time graph. As we said, the slope equals the acceleration, a. And in this graph, the y-intercept is v0. Thus,  
  .
But what if the velocity is not constant? Let’s look back at our jet-car example. At the beginning of the motion, as the car is speeding up, we saw that its position is a curve, as shown in .
You do not have to do this, but you could, theoretically, take the instantaneous velocity at each point on this graph. If you did, you would get , which is just a straight line with a positive slope.
Again, if we take the slope of the velocity vs. time graph, we get the acceleration, the rate of change of the velocity. And, if we take the area under the slope, we get back to the displacement.

### Solving Problems using Velocity–Time Graphs

Most velocity vs. time graphs will be straight lines. When this is the case, our calculations are fairly simple.
Most of the velocity vs. time graphs we will look at will be simple to interpret. Occasionally, we will look at curved graphs of velocity vs. time. More often, these curved graphs occur when something is speeding up, often from rest. Let’s look back at a more realistic velocity vs. time graph of the jet car’s motion that takes this speeding up stage into account.

### Practice Problems


### Check Your Understanding


### Section Summary

1. The slope of a velocity vs. time graph is the acceleration.
2. The area under a velocity vs. time curve is the displacement.
3. Average velocity can be found in a velocity vs. time graph by taking the weighted average of all the velocities.

### Key Equations


### Concept Items


### Critical Thinking


### Problems


### Performance Task


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
