Example: Kinematic graphs and consistency

Understanding the situation

Read the webpages Kinematic Graphs and Kinematic graphs and consistency To see how to think about the different kinematics graphs we can draw and how they relate, consider the following problem.

Presenting a sample problem

A small toy car can move along a horizontal track as shown in the figure below.

Its position is measured by a sonic ranger. When the motion detector is turned on the car is moving towards the left and is speeding up at a uniform rate.

Which of the graphs in the figure at the right can correctly describe what would be shown on the screen if we we chose to plot:

1. 1. A. the position
   2. B. the velocity
   3. C. the acceleration

If none are correct, choose N.

Could the graphs you chose be consistent? That is, could they all describe the same motion?

Solving this problem

What we have to do here is make the connection between our physical picture of the motion and a representation of that motion in the graph. The best way to do this is to first think about the motion and make a mental video of what it would look like. Then, we can run that video in our mind and while we are watching it (possibly in slo-mo!) we read off what we think is happening to each of the variables. Then we consider how each variable will be represented in a graph.

To see how a variable is represented in a graph, we need to understand what coordinate system is being used. Here, nothing is said about what coordinate system to use, so we have to infer what seems most natural given the physical system creating the graph. 

What a sonic ranger typically reports is the distance an object is away from it -- a positive number. That means in this case to the left is positive. Since it reports the distance the object is away from it, 0 must correspond to the car being right at the ranger. (We know it actually can't do that -- it only reports correctly when the object is a some centimeters away, so our position graph cannot start at 0.) We then have to decide how the graphs are representing the measurements. The 0 on each graph indicates that the axes cross when each of the variables being plotted is 0 and the little arrowhead on each axis indicates the positive direction. (Note: In some elementary school texts, an arrow on an axis is used to mean "goes on forever". Since nothing in physics can go on forever, we do not use this convention here.)

Although the graphs are asked for in a particular order, we don't have to do them in that order. It's convenient to choose a starting point that works best for us and matches best the conditions of the problem. In this case. it says the car is "speeding up at a uniform rate." Since it tells us about speed, let's start by figuring out the velocity graph.

First, we note that it says "speeding up" -- so this means that the velocity is not constant but its magnitude is growing. This rules out graphs 1, 2, and 3, which are each constant. Next, let's consider whether the value of the velocity is positive or negative. It says "speeding up" but that doesn't say anything about the direction and the sign of the velocity tells the direction. Since we've decided from the way the ranger works that to the left is the positive direction and since it says it's moving to the left, the velocity should be positive and remain so, since if it's speeding up it won't turn around. This rules out graphs 5 and 7. Then, we note that it says it is "speeding up" so it should go from positive to even more positive. This rules out graphs 6 and 9. We are left with 4 and 8. Finally, it tells us that it is not just speeding up, but "speeding up at a constant rate". That means the rate of change of the velocity (derivative) is a constant. If the rate of change of a function is a constant, then the function is a straight line. This leaves us with graph 4.  Since we could have chosen N (no graph works), let's check that is matches with our mental picture. I see it moving to the left and going faster and faster as time increases. Since it's going in the positive direction it should be going up and at a constant slope. Check!

Now that we have the velocity, we can easily get the acceleration since a = dv/dt. This tells us that the acceleration is just the slope of the velocity graph. Since the slope of the velocity graph doesn't change, the acceleration will be constant. This means it has to be either graph 1, 2, or 3. Since the velocity is increasing (getting more positive) the change in the velocity is positive and the acceleration will be positive. This means graph 1. Since we could have chosen N (no graph works), Let's check that is matches with our mental picture. I see it moving to the left and going faster and faster as time increases. If its speed is increasing constantly, the acceleration graph should be a positive constant. Check!

Lastly, we have to figure out the position graph. Our mental picture tells us that it is starting at a positive value and moving to a larger and larger positive value. Only graphs 4 and 8 do that. Could it be either? If it were increasing its distance from the origin at a constant rate as in graph 4, the rate of change of the position (v = dx/dt) would be a constant, but we know that it is speeding up. So graph 4 doesn't work. Does graph 8? We can read off its velocities by looking at the slopes. Since in graph 8 the graph curves up, the slopes get larger and larger with time so graph 8 is describing a car that is speeding up. (We can use a little math here. We know that if the velocity is increasing at a constant rate it is described by an equation that looks linear in time: something like v = v₀ + at. The function x has to have its derivative look like this. What function of t has df/dt = t ? You should recognize that the answer is ½t ². So the position graph should look like a parabola. Check!) Note that we would have had to reject graph 8 if it started at 0 by the way the sonic ranger works!

So our answers seem to be:

A. the position: 8

B. the velocity: 4

C. the acceleration: 1

We are almost guaranteed to have consistency among our three graphs (that is, they all correctly describe exactly the same motion) but not quite. We constructed them by checking the slopes from each other. But we also have to check that the values are consistent. Even though we noticed that the position graph starts at a positive value (and not at 0), we also have to check that the initial values match. We can read the initial velocity off the velocity graph! Looking at this closely, it looks as if the curve for the position graph comes in flat -- that is, at 0 slope. This means that for graph 8, it is describing the motion of a car that is moving to the left and speeding up, but it is describing one that starts at t=0 with 0 velocity. Graph 4 does not start with 0 velocity so those two graphs are not consistent.

Joe Redish 12/27/14