Example: Reading the physics in a graph

Understanding the situation

A simple example that shows how different graphs can help us make sense of a physical situation by looking at it in different ways is to consider a ball rolling on a track over a hill. In a sense, this relies on the basic physics of position and velocity as described in  the page Kinematic variables, though all you really need is an everyday sense of speed and what happens.

Presenting a sample problem

Consider a ball rolling on a track over a hill. A sketch is shown below. The horizontal position of the ball is described by the coordinate "x" shown underneath the diagram. (Note the arrowhead denoting which end of the x axis is the positive side.)

We'll assume a highly simplified situation. The ball is rolling smoothly, and we can ignore any resistive forces that might be slowing it down. That is, over the distance we are looking at, if there were no hill, the ball would continue at a constant speed (at far as we can tell). We'll assume there are no significant resistive forces; so that after the hill, the ball has returned to the same speed that it started with. Draw graphs of the position and velocity of the ball as a function of time.

Solving this problem

Here's the story of the event. When we start watching (when $t = 0$), the ball is a bit to the left of the origin and is moving at a constant speed. When the ball gets to the hill, we expect it to slow down as it climbs the hill, then continue at a constant slower speed along the flat. When it gets to the end of the flat portion of the hill, it starts downward and speeds up. When it gets to the flat part on the right, we expect it will again continue at a constant speed. Although you may not yet know the physics principles that tell you this, we might guess that whatever speed it lost going up the hill, it gained back coming down the hill. (This is correct given our assumptions.)

Now let's see how to represent this story in two different graphs.

Position and velocity glasses

On the left, we've "put on our position glasses" and have plotted where the ball is along the x-coordinate as a function of time. We've marked five regions of time to match the graphs with our story. On a position-time graph, when an object is moving with a constant velocity, it is changing its position at a constant rate. The time graph of that position is a tilted straight line with a larger tilt representing a larger velocity. Let's read what the graph tells us.

Region I describes from the start of our story until the ball reaches the hill. It starts at t = 0 a bit to the left of the origin so it starts at a negative value. It's moving at a constant speed when we start looking so the graph in region I is a tilted straight line, increasing its position coordinate at a constant rate. Region II describes the times when the ball is rolling uphill. We know it's slowing down so it's curving down a bit. Region III describes the times when the ball is rolling on the flat part at the top of the hill. It's moving at a slower constant velocity so it's again a tilted straight line, but it's not tilted as sharply. Region IV describes the times when the ball is coming down the hill. It's speeding up so its tilt is tilting more upward. In Region V, the ball is again moving at a constant speed — and the same speed we started with. So its graph is a straight line tilted upward with the same slope as in region I.

Now let's take off our position glasses and put on our velocity glasses — the graph at the right.

The velocity graph represents the different regions very differently from the position graph! In this case, we focus on the speed of the ball and how its changing. In region I, the ball moves with a constant positive speed so we have a constant positive graph. In region II, the ball is slowing down so the value of the velocity gets smaller. (We don't know how without applying some physics principles, but for a constant slope of the hill a constant slope on the velocity graph seems like a good guess.) In region III, the ball is moving at a constant speed, but smaller than the one it started at. In region IV, it speeds up again; and in region V, it is again going at a constant speed equal to the one it started at. 

Notice how different these two graphs look! Just compare region I. The position graph tells you the value of the coordinate is continually increasing. You're focusing on where it is. The velocity graph tells you the value of the velocity is not changing. You're focusing on how fast it's going. The follow-ons show (reminds you) how to construct the other graph if you only have one — to infer the velocity graph from the position graph and vice versa.

Joe Redish 12/27/17