The Physics Teacher Vol. 52, 2014

Fermi QuestionsLarry Weinstein, Column EditorOld Dominion University, Norfolk, VA 23529; weinstein@odu.edu.

w Question 1: Charging by walkingCan you put a generator in your shoe and charge your cell phone by walking around? (Thanks to Mariam Abdelhamid of Old Dominion University for suggesting the question.)

Answer: To answer this question, we need to estimate the energy stored in a typical cell-phone battery and the power we can generate while walking. A cell-phone bat-tery has a mass of about 50 g (more than 10 g and less than 250 g). We can estimate the energy density of a bat-tery if we remember that batteries have about 10-2 of the energy density of gasoline and gasoline has an energy density of 53107 J/kg (see Guesstimation1 for details). If perchance we dont remember such arcane facts and numbers, then we can estimate that an incandescent-bulb flashlight powered by two D-batteries can shine for several hours (more than one and less than 24) and use a few watts (more than one and less than 10). The flash-light thus uses

E = (3 W)(5 hr)(43103 s/hr)= 63104 J.

There are about five D-batteries in a kg, so the energy density of a battery is

eD = E/m = (33104 J/batt)(5 batt/kg) =23105 J/kg.

Modern lithium-ion batteries will be better than that, but not a lot better, so we will estimate that

eLi = 43105 J/kg.If our cell-phone battery has a mass of 50 g, then our battery contains

E = eLim = (43105 J/kg)(5310-2 kg) = 33104 J.

According to Wikipedia, that modern source of all non-controversial knowledge, an iPhone battery contains 5.45 Wh, or

E = 5 Wh = (5 W)(43103 s)= 23104 J,

which is embarrassingly close to our estimate.

Now we need to estimate the energy we can supply by walking. This will be the product of the energy we con-sume while walking, the efficiency of converting food

Solutions for Fermi Questions, December 2014

energy (calories) to mechanical energy, and the fraction of that energy that we can divert to electric power pro-duction. My maximum power production, as measured by bicycle-powered light-bulb exhibits at the science museum, is 100 W. Since most people amble rather than power-walk, we will estimate our walking power at 10 W (more than one and less than 100 W). Alterna- tively, if you remember that walking consumes about 100 Calories per hour (more than 10 and less than 1000), then that equals

P = (102 C/hr)(43103 J/C)/(43103 s/hr) = 102 W.

However, that is the chemical energy (i.e., food) con-sumed, not the mechanical power used to walk. The power output will be a fraction of the chemical energy consumed, or about 30 W (more than 10% and less than 100%). Averaging the two, well estimate that walking uses 20 W of mechanical power.

How much of this power can we divert to generate elec-tricity? If we divert too much, then it will feel like were walking on really soft sand. We can divert more than 1% and less than 100% so well estimate 10%, or 2 W.

At that rate, we will recharge our batteries in a mere 104 s or 3 hours. If we treat our legs as simple pendula, then each step takes a time

and we will take 104 steps in those 104 s. Coincidentally, this is exactly the goal of many activity routines, such as fitbit.

Now we just need a small, efficient, piezo-electric generator in our shoes and our phones will always be charged!

1. Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, by Lawrence Weinstein and John Adam (Princeton University Press, 2008).

Copyright 2014, Lawrence Weinstein.

The Physics Teacher Vol. 52, December 2014

w Question 2: Saving daylightAs the winter solstice approaches, daylight is becoming scarcer and more valuable. How much more daylight could we enjoy by driving west during the day and east at night?

Answer: The amount of extra daylight we could enjoy would depend on our driving speed and the Earths rotational speed. This problem is one of the few that is actually easier in U.S. customary units. The Earths rota-tional speed is

at the equator. (In other words, each time zone is 103 mi across at the equator.) Rome and New York are both at about latitude 40o N so that their rotational speed is

(or 103 km/hr for those readers who dont appreciate the quaint charm of U.S. customary units or 33102 m/s for purists who insist on SI units).

If we can find a nice uncluttered east-west interstate or autobahn, we can easily drive at 70 mph (or 100 kph or 30 m/s [not that there are any automobile speedometers marked in m/s]). This is 10% of the Earths rotational speed, so that for each unit of time spent driving west during the day, we will gain 0.1 units of time of day-light. If we drive one hour west every day and one hour east every night, we will enjoy six extra minutes of day-light. It hardly seems worth it.

On the other hand, I work about 20 miles west of where I live. In December I drive west after sunrise and east after sunset. This means that I gain an extra

of daylight every day for free.

Oh boy.

Copyright 2014, Lawrence Weinstein.