2017 AP Physics 1 problem 1 with the draining batteries: we set it up

On the last day of my AP Physics 1 summer institutes, participants are asked to pick a released exam question, solve it... and then set it up experimentally.  We always get some simple yet elegant setups, like Rebecca’s modified-Atwood-turned-projectile from the 1998 exam; some complicated setups that don’t quite work but point in the right direction, like Lorren’s attempt at the 2018 collision-on-a-horizontal-spring problem which got us thinking about vertical springs and velcro; and some complex and elegant creations that go beyond just the AP question, like Quinn’s measurement of the kinetic friction coefficient that can then be used to determine the mass of an unknown block.

Several free response problems over the last few years have presented significant experimental challenges, though.  When the bumpy track problem came out on the 2016 exam I heard from so many teachers, "this isn't real, no one would actually set that up.  Well, we did last summer. 

This year one participant, Frank, took on an even greater challenge: the paragraph question about draining batteries, from the 2017 AP Physics exam.  It postulates connecting light bulbs to a battery in three ways, and asks which will drain the battery soonest.  

Which battery drains first depends on the power dissipated by the circuit: since power is energy per time, the shortest time to drain the battery will come from the circuit dissipating the largest power. By V²/R with the battery always providing the same voltage, whichever circuit has the smallest equivalent resistance will dissipate the most power.  So the bulbs in parallel will drain the battery first, followed by the single bulb, followed by the two bulbs in series.  That's the theoretical solution.

Frank and I and our lab assistant Tom brainstormed three approaches to this experiment.

(1) We could finesse the problem by simply observing brightness, which correlates directly to power.  That's almost trivial - we connected three sets of bulbs to three identical batteries, and we saw the parallel bulbs brightest, the single bulb next brightest, and the series bulbs dimmest.  Great for a simple demo, but this sort of thing has been done before.    

(2) We could get three fresh batteries, hook them up, and set up an iphone to do a time lapse video.  I don't know offhand how long it would take to drain fresh AA batteries, but it's not gonna happen within a single lab period; and that's all we had at our institute.  I do hope that some reader will set up the time lapse and send it along.

(3) Tom noted that he had some "big blue" capacitors available.  These are 27,000 microfarads and can operate up to 25 V.  Though the study of capacitors is beyond the scope of AP Physics 1, it's not hard to explain to students that these things store a fixed amount of energy (for a given starting voltage), and then that energy dissipates rapidly.  

How rapidly?  Your students don't need to know, but the time constant of discharge depends on the product of the equivalent resistance and the capacitance.  We want an equivalent resistance in the dozens of ohms to get time constants on the order of seconds.  Frank and Tom found some Christmas lights; when these were connected via alligator clips, the dimming of the lights over a few seconds as the capacitor drained was easily visible.

Which drained first?  It was sorta hard to tell with the naked eye.  So we clothed our eyes with slow motion video on our phones.  

Frank charged the capacitors.  He connected a switch that, when thrown, would simultaneously connect each circuit to its capacitor.  Here's the result:

Just like we predicted - physics works.