Kinematic graphs and consistency


We've now defined three kinematic variables: position, velocity, and acceleration. (We could keep going with our chain of derivatives: x(t), v(t) = dx/dt, a(t) = dv/dt, ..., but for most of what we'll be interested in, that's enough.) For any single motion of an object, we can now draw three graphs as a function of time: a position graph (x(t) vs t), a velocity graph (v(t) vs t), and an acceleration graph (a(t) vs t). These three graphs each describe the motion and present information about it. Each of these are "graphs for the mind" -- they have to be interpreted and don't look like what we see with our eyes. (See Kinematics graphs on "graphs for the eye and graphs for the mind".)

Here's an example. We took the video of a juggler and tracked the central ball as it went up and down. Here are the graphs that the video capture program LoggerPro™ produced for position ($y$), velocity ($v$), and acceleration ($a$).

Although each of these three graphs represent the same motion, they look very different and they each tell us something different about the motion. As the course goes on, we will add more variables that describe the motion -- forces, momenta, energies -- and learning to work with multiple graphical representations will become increasingly important. This is a common technique in science. Often, the same phenomena is presented graphically in multiple ways. with each graph making some characteristic of the phenomenon more salient. 

The interesting and important point is consistency. If these three graphs are supposed to represent the same physical motion, then there are relations among them that have to be satisfied.  This is our safety net of cross-links that tell us that we are doing the right thing.

For these three graphs -- position, velocity, and acceleration -- here are a number of different and useful ways of checking the cross links.

  • Make a mental picture of the physical system -- For motion problems this is almost always the most useful first step. Imagine a video of the system moving and then run it in your mind multiple times looking for different things. Where is it? How fast is it moving and in what direction? How is the velocity changing? Having a single mental movie that you can read in different ways is one of the most valuable ways to guarantee consistency.
  • Do the math to check the slopes --  In the case of position, velocity, and acceleration, the graphs are mathematically related. Velocity is the derivative of position and acceleration is the derivative of velocity. You can look at the curves and fit them with a function (in the example above t2, t, and constant respectively) and see that the functions are derivatives of each other). Or you can do the "little triangle trick" and slide a small triangle along each curve, getting a sense of how the slope changes to see what the derivative curve should look like, going from position to velocity to acceleration.
  • Do the math to check the areas -- Or you can run it backwards -- from acceleration to velocity to position -- by doing the integral by looking at how the area under each curve. 
  • Check the coordinate system and starting values -- One place that students often slip up in reading graphs is making assumptions about the choice of coordinate system and origins, both for the space and time variables. If you've had high-school physics, you might have gotten used to the idea that there is a "standard" coordinate system -- with x being horizontal and pointing right, y being vertical and pointing up, and the origins of both space and time being the starting point. While this is convenient when you are getting started, it's too restrictive for more complex problems. We will often NOT stick to these conventions -- and we'll often shift them to give you practice. Watch out!

For analysis of how this might work in a problem, check out the example in the Follow-on.

Joe Redish 12/28/14 


Article 329
Last Modified: March 7, 2019