Honors Physics - Graphing Motion

Position-Time (x-t) Graphs

Particle diagrams can help you understand an object's motion, but they don't always tell you the whole story. You'll have to investigate some other types of motion graphs to get a clearer picture.

The position-time graph shows the displacement (or, in the case of scalar quantities, distance) of an object as a function of time.  Positive displacements indicate the object's position is in the positive direction from its starting point, while negative displacements indicate the object's position is opposite the positive direction.  Let's look at a few examples.

Suppose Cricket the WonderDog wanders away from her house at a constant velocity of 1 m/s, stopping only when she's 5m away (which, of course, takes 5 seconds). She then decides to take a short five-second rest in the grass. After her five second rest, she hears the dinner bell ring, so she runs back to the house at a speed of 2 m/s. The displacement-time graph for her motion would look something like this:

As you can see from the plot, Cricket's displacement begins at zero meters at time zero. Then, as time progresses, Cricket's displacement increases at a rate of 1 m/s, so that after one second, Cricket is one meter away from her starting point. After two seconds, she's two meters away, and so forth, until she reaches her maximum displacement of five meters from her starting point at a time of five seconds. Cricket then remains at that position for 5 seconds while she takes a rest. Following her rest, at time t=10 seconds, Cricket hears the dinner bell and races back to the house at a speed of 2 m/s, so the graph ends when Cricket returns to her starting point at the house, a total distance traveled of 10m, and a total displacement of zero meters.

As you look at the position-time graph, notice that at the beginning, when Cricket is moving in a positive direction, the graph has a positive slope. When the graph is flat (has a zero slope) Cricket is not moving. And when the graph has a negative slope, Cricket is moving in the negative direction. It's also easy to see that the steeper the slope of the graph, the faster Cricket is moving.

Question: The graph below represents the displacement of an object moving in a straight line as a function of time. What was the total distance traveled by the object during the 10-second time interval?

 

Answer: Total distance traveled is 8 meters forward from 0 to 4 seconds, then 8 meters forward from 6 to 8 seconds, then 8 meters backward from 8 to 10 seconds, for a total of 24 meters.

 

Velocity-Time (v-t) Graphs

Just as important to understanding motion is the velocity-time graph, which shows the velocity of an object on the y-axis, and time on the x-axis. Positive values indicate velocities in the positive direction, while negative values indicate velocities in the opposite direction. In reading these graphs, it’s important to realize that a straight horizontal line indicates the object maintaining a constant velocity – it can still be moving, its velocity just isn’t changing. A value of 0 on the v-t graph indicates the object has come to a stop. If the graph crosses the x-axis, the object was moving in one direction, came to a stop, and switched the direction of its motion. Let’s look at the v-t graph for Cricket the Wonderdog’s Adventure:

For the first five seconds of Cricket’s journey, you can see she maintains a constant velocity of 1 m/s. Then, when she stops to rest, her velocity changes to zero for the duration of her rest. Finally, when she races back to the house for dinner, she maintains a negative velocity of 2 m/s. Because velocity is a vector, the negative sign indicates that Cricket’s velocity is in the opposite direction (initially the direction away from the house was positive, so back toward the house must be negative!)

As I’m sure you can imagine, the position-time graph of an object’s motion and the v-t graph of an object’s motion are closely related. You’ll explore these relationships next.

 

Graph Transformations

In looking at a position-time graph, the faster an object’s position/displacement changes, the steeper the slope of the line. Since velocity is the rate at which an object’s position changes, the slope of the position-time graph at any given point in time gives you the velocity at that point in time. You can obtain the slope of the position-time graph using the following formula:

Realizing that the rise in the graph is actually ∆x, and the run is ∆t, you can substitute these variables into the slope equation to find:

With a little bit of interpretation, it’s easy to show that the slope is really just change in position over time, which is the definition of velocity. Put directly, the slope of the position-time graph gives you the velocity.

Of course, it only makes sense that if you can determine velocity from the position-time graph, you should be able to work backward to determine change in position (displacement) from the v-t graph. If you have a v-t graph, and you want to know how much an object’s position changed in a time interval, take the area under the curve within that time interval.

So, if taking the slope of the position-time graph gives you the rate of change of position, which is called velocity, what do you get when you take the slope of the v-t graph? You get the rate of change of velocity, which is called acceleration! The slope of the v-t graph, therefore, tells you the acceleration of an object.

Let's take a look at a sample problem from the June 2009 Regents Physics Exam:

Now that you've seen how to solve these types of problems, why don't you try a few on your own?

Acceleration-Time (a-t) Graphs

Much like velocity, you can make a graph of acceleration vs. time by plotting the rate of change of an object’s velocity (its acceleration) on the y-axis, and placing time on the x-axis.

When you took the slope of the position-time graph, you obtained the object’s velocity. In the same way, taking the slope of the v-t graph gives you the object’s acceleration. Going the other direction, when you analyzed the v-t graph, you found that taking the area under the v-t graph provided you with information about the object’s change in position. In similar fashion, taking the area under the a-t graph tells you how much an object’s velocity changes.

Putting it all together, you can go from position-time to velocity-time by taking the slope, and you can go from velocity-time to acceleration-time by taking the slope. Or, going the other direction, the area under the acceleration-time curve gives you an object’s change in velocity, and the area under the velocity-time curve gives you an object’s change in position.

Let's take a look at an example problem to demonstrate this concept.