# Calories used running

1. Dec 6, 2007

### pman

Hello:

I am trying to figure out a way to calculate the number of calories burnt running. Before I do on to explain how I am doing this, I realize that there are many calculators out there that will do this for me, my interest is in finding out how this is calculated. I have had some conversations with a friend of mine that knows enough on this subject and his argument is that it does not matter how fast you are moving, what matter is how much you weight and the distance you travelled. In other words by his take, you would burn the same amount of calories by walking a mile in 60 minutes or running that mile in 12 minutes.

I had read many posts that support either side of this issue, some agree with the statement above others say that it is innacurate and that straining yourself, accelerates your heart rate (thus your metabolism) and you end up burning more calories in the end.

So i thought I'd try to figure out the physics behind the issue. These are the formulas I am using:

Work (joules) = Force (N) * Distance (m)
Force (N) = Acceleration (m/s^2) * Mass (Kg)
Acceleration (m/s^2) = (2 * Distance) / Time^2 (s)

These are the units of convertion I am using
1 Mile ~= 1609 Meters
1 calorie ~= 4.18400 joules
1 pound = 0.45 Kg

This is a sample calculation.

Test Weight = 200lbs = 90.71Kg
Distance Travelled = 1 Mile = 1609 meters
Time spent = 31 minutes = 1860 seconds

Acceleration = 3218.688 (m) / 3459600 (s^2) = 0.000930364 (m/s^2)
Force = 0.000930364 (m/s^2) * 90.7184 (Kg) = 0.084401152 (N)
Work = 0.084401152 (N) * 1609.344 (m) = 135.8304876 (J)

Calories = 135.8304876 / 4.18400 = 32.46426565

Ok so, with those numbers if the calculations are correct, i could say that if I ran for 31 minutes at any speed rate that would amount the distance to a mile I would burn 32.46 calories. However, if i keep everything else the same except that this time I took 60 minutes to reach the mile (more like a slow walk). Then my numbers are:

Acceleration = 3218.688 (m) / 12960000 (s^2) = 0.000248356 (m/s^2)
Force = 0.000248356 (m/s^2) * 90.7184 (Kg) = 0.022530419 (N)
Work = 0.022530419 (N) * 1609.344 (m) = 36.25919404 (J)

Calories = 36.25919404 / 4.18400 = 8.666155

Does this disprove that the whole concept of weight & distance will constantly give you the same results as far as calories used regardless of speed travelled? or are my calculations incorrect?

My friend pointed me to this calculator http://www.bodybuilding.com/fun/calories.htm and although they use speed to calculate calories burned (probably used to calculate distance) you can see that if you enter in 200lbs as the weight and 60 minutes as the time (select popular cardio exercises) in the results page: it says that running 10 mph (6 min/mile) burns 1524 calories and running 5 mph (12 min/mile) burns 762 calories, and it makes sense since at @5mph you cover half of the distance. My two questions here are how come my calculations do not reflect that? should't travelling a mile in 60 minutes only half the 32 calories burn at 31 minutes and produce a result of 16? Also I've tried to compare the same numbers as bodybuilding.com for instance running 5 miles in 60 minutes, and my numbers come back as only 216 calories used but bb.com comes back as 762. I also just noticed that using the bodybuilding.com calculator with the same numbers as before (200lbs and 60minutes) walking, @2.5 mph, burns 286 but running 5 mph (12 min/mile) burns 762 calories since the distance covered walking is half of that covered when running, shouldn't the calories burned be 381? Is there a different way to calculate all of this that does not include the formulas above?

Any help at all would be appreciatted. Thanx!

Last edited by a moderator: Apr 23, 2017
2. Dec 6, 2007

### f95toli

There is no simple way to calculate this.
There is a whole field called biomechanics where they study "energy consumption" etc.
The amount of calories you need to burn in order to run a certain distance basically depends on what is called "running economy"; which in turn depends on your running form (how you move your limbs), what type of muscles you have (which is mostly genetic), your weight, how fast you are running ("all-out" running is very different from jogging) etc. The kind of calculators you find online essentially use empircal formulas; for e.g. an elite long-distance runner the real numbers could be very different (because they tend to have MUCH better running economy than your "average" jogger).

Also, I think there is something fishy with your calculation. Most of the energy is used to counteract gravity, i.e. to "lift" the body each step. You seem to be assuming that the runner is accelerating the whole distance which is clearly not true.

3. Dec 6, 2007

### Office_Shredder

Staff Emeritus
There are two problems with your calculation:
1.) You don't take into account friction
2.) You assume that the person accelerates the whole time

Basically, instead of calculating how much work it takes to travel against friction (as it should DEFINITELY be noted that the act of moving side to side at a constant velocity in a gravity field takes no energy) you calculated some phantom number that's meaningless

4. Dec 6, 2007

### chroot

Staff Emeritus
You're not going to be able to calculate this from first principles -- there are hundreds of different variables involved. You're going to have to settle for simply measuring it in an experiment.

The most common way of measuring calorie expenditure involves measuring heart rate. It turns out that oxygen consumption (and thus energy expenditure) are relatively tightly correlated with heart rate. If you know your heart rate, you know within perhaps 20% error how many calories you're burning per hour. You can buy heart rate monitors from sporting goods stores for very modest prices.

- Warren

5. Dec 6, 2007

### FredGarvin

Oxygen intake is a major factor that is measured in any exercise study. Like it has already been said, you can not get an accurate calculation based on basic mechanics.

6. Dec 7, 2007

### pman

Thanx for your answers guys. This helped to clear some of my misconceptions!

7. Dec 7, 2007

### f95toli

Perhaps Slightly OT...
It is interesting to note that the amount of energy needed to run is actually quite small. We are very efficient runners compared to most other animals. There are plenty of animals that can outrun us over short distances but AFAIK a reasonably fit human can outrun just about every other land based animal over long distances.
Even an average (healthy) human can relatively quickly (less than a year) reach a fitness level that allow him/her to run for 4-5 hours without stopping (just think of the people participating in marathons, most of them are not "superhuman").
Some evoutionary biologists believe that we evolved to become natural long distance (i.e. endurance) runners, perhaps because it made it possible to hunt rather large animals by pursuing them until they collapsed due to exhaustion (which can take many hours if you are hunting something like a deer) . This technique is still used by some tribes in Africa.
There was a paper about his in Nature quite recently.