Apropos of the calorie/Calorie discussion yesterday, here’s something interesting to think about with regard to the energy used in exercise.

The formula for gravitational potential energy is m*g*h, where m is mass, g is the acceleration due to gravity, and h is the height. Of course this is if the changes in h are small enough so that g can be taken as truly constant. So any change in height is going to require energy if you’re going up, and will release energy if you’re going down. This is why you hit the gas driving up a hill and hit the brake going down a hill. Actually since we’re talking exercise let’s change that to biking. Uphill is hard, downhill is easy.

Now if you’re going a long way uphill your height changes quite a bit, and correspondingly so does your potential energy. Should you decide to climb the stairs of the Empire State Building you will ascend some 1,050 feet. Now let’s say you weigh 160 pounds and that the gravitational acceleration is the usual 9.8 meters per second squared. We’re after mgh, so multiply them all together after of course converting everything to SI units. I get a total change in potential energy of roughly 230,000 joules. That works out to 54 food calories.

54! Surely something that difficult would burn a lot more calories, you’d think. And it does. The immense effort you expend in climbing is mostly budgeted to different bodily processes. You have to move extra air in and out of your lungs. You have to circulate blood at a much higher rate. You have to process the complicated chemistry required to keep your muscles moving. All of these things take energy, and by the time the shoe meets the stair most of the energy has already been lost, eventually ending up mostly in the form of heat. Your body can’t afford to overheat and so you begin sweating to carry the excess heat energy away. All that energy had to come from somewhere, and it came from the food you ate. By the time you’re on the observation deck looking over Manhattan you’ll have used up a lot more than 54 calories.

I like to spring this as a quiz question on my 201 students near the beginning of the semester. One of the most important things in physics is developing an intuition for roughly what the right answer should be, so you can tell if you’ve made an obvious mistake somewhere. But for this to work you have to have worked out for yourself how these physics problems look in terms of real life quantities. Sometimes it’s not what you’d expect, and this is one of those cases.