Diet plans, Coca-Cola and stair climbing

I just found a photo of a can of coke (the real stuff, not that vile diet swill) next to a pile of the ten sugar cubes, supposedly dissolved in it. Gee! Ten cubes? That much? Boy, my health insurance must go berserk if they knew, I drank this. I mean it's ten cubes, that's ... well ... it's ... uhm ... shocking, I suppose... I think, I am totally worried for my health here! That much glucose can't possibly be good for me, right? It's a miracle, I don't drop dead from just looking at it!

Ok, enough now with the diet industry scaremongering, how bad is it really?

First of all, sugar is not unhealthy by definition. In fact, we even depend on carbon hydrates in our diet as our primary energy source. The only problem is that before modern technology revolutionized agriculture, our food contained more fiber than sugar, forcing our bodies to evolve to process "large" quantities of food in order to get enough nutrients. Now, it's almost the other way around, resulting in us eating more calories than we need.
In principle, a high carb diet is not nescessarily a problem (at least not unless you are a diabetic). You just have to work it off if you do not want your body to store the surplus energy as fat.

So, back to our scary ten cubes per can. How much physical workout is actually needed to burn away those calories? Let's set our mind on using the oldest and cheapest fitness machine, gym owners don't want you to know about: stairs.
How many stairs do you have to climb after drinking one glass of coca cola?

Before doing the math, let's do a small (thought) experiment. In honor of Sir Isaac Newton (and your health), place an apple on the edge of your desk, tip it over and measure the time, it takes to fall to the ground. Now, pick the apple up again and repeat that experiment two times: the first time by picking the apple up fast, the second by picking it up very slowly. Does the speed of you picking the apple up affect the speed by which it is falling down? The answer should be: "no" (if it does, contact your local physicist - you are on to something).
Now, this is an important observation. According to the law of conservation of matter and energy, a falling object releases exactly the same amount of energy that was previously stored in it by lifting it up. Since the apple always falls at the same speed, obviously the amount of energy stored is independent of the speed by which it was put on the desk.
What does this mean? Well, two things:

  1. Whether you walk or sprint the stairs up requires the same amount of energy.
  2. Instead of climbing up the stairs, we can also sneakily fall them down, calculate the energy released and then simply claim this to be what we previously invested.

So, how much energy does a can of coke contain? Coca-cola states in the nutrition facts table of their bottles that 100 ml of coke (about one third the contents of a can) contain 10.6 g sugar, which is about 180 kJ of energy (mind the "kilo" here). One Joule is defined as:

Ok, that's one scary equation, I know. Let's try to make it look friendlier:

Much better. You may remember the rightmost part from driving school. where it is used to calculate the fuel consumption for an accelerating a car. Let's transform the equation one more time to bring meter to the left hand side:

This finally looks usable. All we need now are some numbers to plug into it.
From the coke can, we know that we have to burn 180 kJ and hopefully, you also know your own weight. But what exactly is a suppose to be and more importantly, how big is it? In physics, a means acceleration. The acceleration, we are talking about here is caused by earth's gravity (remember: we are calculating the energy consumption of climbing stairs by actually falling them down instead) and this is a constant value of about 10 m/s2 (or 9.81 Nm to be more precise).

Now, let's assume, you weight 60 kg and drank 100 ml of Coca-cola. How many stairs would you have to climb up?

Err, 300 meters altitude difference? Drinking 100 ml of coke enables you to climb up a skyscraper? That can't possibly be correct. There must be some mistake here somewhere, right? Right! The model above actually applies to a 100% energy efficient crane engine, lifting a sack of potatoes. A human body is neither a sack of potatoes nor 100% percent energy efficient:

  • While at rest, we spend about 100 J/s on producing body heat (more when exercising)
  • When climbing stairs, we usually don't just move our legs, but our arms as well. Moving your arms burns calories, but does not nescessarily move your forwards (unless you pull yourself along the handrail).
  • Heart, brain and other internal organs also constantly spend energy.
  • Our muscles are far from being energy efficient. In fact, their efficiency factor is only about 25-30%. Meaning, only 25 out of one hundred calories are used for movement, while the remaining three quarters heat up the muscle or are spend on chemical reactions.

For simplicities sake, let's ignore the entire body upkeep and only use the efficiency factor of 25% to quarter our calorie intake (we'll still be using the full 180 kJ, but since movement causes our legs to warm up, only 45 kJ are used for actual transportation). This gives us the following, handy, equation:

That's it. On the assumption that you do not sprint (burn more energy through arm/torso movement, increased heart rate and sweating), simply divide 4500 by your current body weight in kilograms (including clothes) and you have the maximum altitude difference in meters, you have to climb for every 100 ml of coke.