Power Calculations

Power is the term we use to describe the rate at which energy is used. Or equivalently, it is the rate at which work is done (in this case, work means mechanical work, gravitational work, etc.).

You hear all the time about power settings on exercise equipment... Set the bike for 200 Watts, and so on. Well, that just means it's going to take a certain amount of energy on your part, spread out over some amount of time.

Mathematically, Power = Energy / Time. So, if you expend 100 Calories of energy over 1 hour of time, you would describe your average power output as 100 Calories per hour. I say "average" here because we're only interested in the TOTAL energy and the TOTAL time, not the instantaneous values (the values at a single moment).

However, Calories per hour is not a standard unit for power. We typically use the Watt to represent power, and just as you might purchase a 100 Watt light bulb, you can do 100 Watts of output on a walk, run, or bike. Or, just sitting around, too. But remember that TIME is part of the equation... the longer you spend expending some fixed amount of energy, the lower your power output would be. You end up dividing by a larger number, since time is in the denominator, so that makes sense.

So the standard unit is the Watt, and that is defined as 1 Joule per second. A Joule is the standard unit of energy. We're more used to thinking in terms of the Calorie, though, as a unit of energy. A banana has 100 Calories of chemical energy stored in it, for example. Well, one Calorie is the same as 4,200 Joules. So if I burn 200 Calories in 2 hours of exercise, then my average power is P = 200 C / 2 h  = 200 C * 4,200 J/C / 2 h * 3,600 s/h = 117 J/s = 117 Watts. As you can see, if I burned the same number of calories, but spread it out over 4 hours of exercise (by exercising less intensely, or by taking lots of breaks, or whatever), then my average power would be P = 58 Watts.

It turns out that lots of physiologists have done studies to figure out how many Calories we burn while walking and running. As you might expect, this depends on your weight... great. It turns out that running burns 0.63 * Weight (in pounds) number of Calories per mile, while walking burns 0.3 * Weight (in pounds).

I should mention that the values above are to calculate your NET Calories burned... meaning, those above and beyond your Base Metabolic Rate (BMR) Calories.

So, if you weight 100 pounds, you'll burn roughly 63 Calories per mile while running, and 30 Calories per mile while walking. This is for "medium" speeds (3-4 mph walking, for example).

Using all this information, and given my current weight of 326 pounds, by doing three walking loops and one running loop (for a total of one mile), I burn 0.75 * (0.3 * 326) + 0.25 * (0.63 * 326) = 125 Calories. Now, the faster I do that mile, the greater my average power output must be. Since I only time myself once a week, and I've only been doing this a week... I'll use my first set of times. My total exercise time was 17:08, or 0.28566 hours.

Therefore, I burn 125 Calories / 0.28566 hours = 436.6 Calories per hour. Converting this to standard units (remember 1 Calorie = 4,200 Joules, and 1 hours = 3,600 seconds) yields: 509 Watts.

My average power output is 509 Watts over the 17 minutes it takes me to do my mile.

What should be obvious is that since I'm capable of maintaining 509 Watts over 17 minutes, as I loose weight, I should be able to see improvements in my times, even if my fitness doesn't improve at all. Simply by weighing less, I could go much faster... If I weighed 200 pounds, I'd only need 77 Calories to go a mile, then my 509 Watts would get me around the course in 10:31. Now, that's still not really a world class mile time... but in reality my fitness (and therefore my ability to generate a larger average power output) will be increasing as well, so by the time I've lost all that weight my average power might be closer to 750 Watts, which would give me a mile time of 7:08. Still not world class, but that would be damn acceptable to me!