Setting up authentic AP problems in lab - 2015 P1 #3

One of the all-time best sources for college-level, open-ended laboratory ideas is the AP exam itself.  While some AP problems are explicitly posed in an experimental setting, the large majority of the released problems - free response and multiple choice - can lead to an interesting laboratory investigation for students in the latter half of the school year.  See this post, this post, this post... you get the idea.

It took me several attempts over the years to get decent data for 2015 AP Physics 1 problem 3.  (Here's the link to the 2015 free response questions.)  To summarize, the question posits a block in contact with a compressed spring on a frictionless surface.  The spring is released from rest.  At the spring's equilibrium position, the block comes off the spring onto a rough surface.

The primary experimental challenge is to produce data verifying the answer to part (a)(i): Sketch a graph of the block's kinetic energy as a function of position.  In particular, we need to show the correct shape of the graph before and after the block reaches the spring's equilibrium position.

The theory: Okay, obviously the kinetic energy increases to a maximum at the D = 0 equilibrium position, then decreases back to zero, because the speed increases than decreases.  The functional form of the graph is the complicated bit.  It's easiest to see through an energy approach:

Before D = 0, the potential energy of the spring is given by (1/2)kD².  This means the potential energy drops to zero as D², i.e. parobolically.  Since the sum of kinetic and potential energy must be a constant value on this frictionless surface, the kinetic energy curve must be an upside-down parabola.

After D = 0, the block loses kinetic energy because work is done on the block by the friction force.  Work is force times distance traveled parallel to the force... that is, linear with distance.  Thus, the kinetic energy drops linearly until the block stops.

The experiment: You'll need a compressible spring, a way of keeping the block moving in a straight line, and a device to record speed as a function of position.

(You DON'T need to worry about the "frictionless" surface before D = 0.  The shape of the graph is still parabolic, even if there's work done by friction before the block is released.  Why?  Because both the with- and without-friction mathematical functions -- (c - kD²) and (c - fD - kD²) -- are parabolic.)

My students and I flipped a PASCO two-meter track upside down to find a groove just about the right size to fit one of those wooden felt-covered friction blocks.  We stuck a Vernier sonic motion detector in the groove about a meter away from the track's edge.  The detector was set to acquire data at the highest possible frequency - I think we used 50 points per second, but I'm not entirely sure.  At the edge of the track, we pushed the block against the spring and let go.  The spring uncompressed, the block slid toward the clicking detector. 

Next came some serious data presentation work.  We had to figure out how to use the Vernier Labquest to graph calculated data columns - it only automatically produces distance, velocity, and acceleration data.  But this is a matter of programming, not physics.  We got the device to output (1/2)mv² on the vertical axis; on the horizontal, we used a subtraction function to adjust the distance from the detector so that "zero" represented the location of the block with the spring uncompressed. 

And sure enough... though the function looks choppy because we only have one point every 0.02 s, it's the correct shape:

As an improvement, a student suggested using a PASCO smart cart on a track, attaching a crumpled piece of paper or similar to the front of the cart to produce some frictional drag on the track.  The motion encoder in the smart cart's wheels can take more frequent (and more precise) data than the sonic sensor.  However, I don't have enough familiarity with the sparkvue interface to create the derived graph from the raw data.  If you do, please try the experiment and email me the results - I'll post them here!