# More on energy: Transforming a modeling problem to gain deeper insights about energy

A comment on my last blog post from Neal reminded me of a problem that is part of the canon of the modeling curriculum:

Another idea related to energy scales and the energy density of gasoline… Try converting “miles per gallon” into units of area, the interpretation being, What if instead of having a tank of gas, my car slurped up the gas it needed as it drove along? How much gas would we need to drizzle along the road? WolframAlpha even provides some nice size comparisons… Shockingly, for a 40mpg car, the cross sectional area of that line of gasoline is on the order of a single pixel, or about 7 times the cross sectional area of a human hair.

I vaguely recall hearing this first as an example Feynman gave in some interview/lecture…

Here’s the question, which comes near the end of the worksheets devoted to introducing the pie chart representation.

First, this is a great problem since it really pushes student’s conception of energy to really think about thermal or dissipated energy, and they have to overcome their first instinct just to draw a pie chart that never changes and is all kinetic energy. When students first present this idea, someone asks “from the look of your pie chart, it seems like the truck could just go on forever…could it?”

Soon they end up drawing this answer:

It’s great that students have figured out that as the truck drives down the road the chemical energy of the gasoline is being transformed into thermal energy, and eventually, when the car runs out of gasoline, the kinetic energy will be transformed to thermal energy too, bringing the truck to a stop. But do they really understand what’s going on here? How quickly is this process occurring? And does it really matter if we’re talking about tractor trailer, truck, car, or motorcycle?

This problem is introduced very early in the energy unit, often before we even have a unit for energy, so I don’t want students to necessarily be able to calculate a quantitative rate at which these transfers are taking place, but what about adding a second question to this problem to probe a little bit more deeply:

I think this might generate even more thinking and discussion and lead to some real insights about energy that might help inform students future choices as energy consumers—something I find lacking in how I treat energy presently.

Finishing it off with this XKDC What if on bird droppings and miles per gallon that Neal referenced would be the icing on the cake.

At first glance, I think the motorcycle’s pie charts would be split in the same proportions: while the motorcycle has less mass (and therefore less kinetic energy at the same speed), it also has a smaller gas tank (AFAIK) and needs fewer gallons per mile. So, I’m not sure it would necessarily generate fruitful discussion without explicitly comparing the two cases. As in some variant of:

7. Which characteristics of your pie charts are the same for both the truck and the motorcycle? Which characteristics of your pie charts for the truck and the motorcycle are different? Relate your answers to the similarities and differences between real trucks and motorcycles.

Of course, this sort of explicit comparison is probably a good idea for some of the other pairs of objects, as well…

Jen,

You raise some great points. But the motorcycle’s engine is surely consuming gasoline at a far smaller rate than a truck’s engine, and therefore the rate at which Uchem is converted to Etherm should be smaller, no?

I think you’re also right that some of the other pie charts need prompting to invite comparison and deeper analysis.