One of my favorite mechanics labs is the classic PSSC lab that uses falling coffee filters to characterize the drag force. I decided to use this as a challenge problem to wrap up the BFPM unit with my honors classes.

The first thing my students did was figure out that the filters were falling at constant velocity using the motion detector. Once they do this, they quickly draw the following free body diagram.

FBD for falling filter. The drag and gravitational forces are equal

Then the students realize you can add a filter to the coffee filter, and suddenly take raft of data before the can really realize what they’re doing. In no time, they have a graph like this

A plot of drag force vs. velocity

As you can see, they’re still in the “graph first, ask questions later” mode, but a few questions reveal they’re really starting to think deeply about physics.

Me: What does the horizontal intercept of this graph tell you.
S: if the filter had that velocty, the drag force would be 0.
Me: and the vertical intercept?
S: That if the speed were zero, somehow the air would exert a force opposite the direction it normally does.
Me: do either of these things make sense?
S: no.
Me: So what if you graph the drag force vs the square of the velocity?
S: Why would you do that? .
Me: try it and see

And so they quickly find this:

A graph of drag force vs the square of the velocity. This shows that the drag force is proportional to the square of the velocity.

S: Hey, this tells us that the drag force and velocity squared are directly related.

Of course, at this point, kids really don’t have any idea what they’ve just discovered. So we dig deeper.In order to get kids to do more proportional reasoning, I ask “hey, when you double the number of filters, doubling the drag force, what happens to the speed?” They see the speed isn’t double, it’s only a small amount larger. “Why?” I ask. Soon, they realize that it must have something to with the relationship they just discovered that the drag force and velocity aren’t directly related, the drag force and the square of the velocity are directly related.

But the big confusion comes when you ask the kids to tell you what the drag force depends on. Inevitably, they say “speed and mass.”

“Yes,” they say ” when you add mass, the weight increases and we see that the drag force is bigger.”

“So if I’m I’m driving down the highway at 55 mi/h and I pick up a passenger, the drag force of the air on my car will increase?” I ask.

“Well, no. But in our experiment, as we added mass, the drag force got bigger.”

About 5 why’s later, the kids turn back to their FBD, and realize that the reason drag force and and weight are equal is only because the filter is traveling at constant velocity, and as the filters get heavier, it takes longer for them to reach a higher terminal velocity. These are the small things I would have probably glossed over a few years ago that really show deep understanding, and end up being the focus of many of our group discussions. It’s awesome to see kids figuring this out together, and then working to make sure every student gets it.