Physics Question: Who gave more effort? Calling all Numbers Geeks!

E/kg wrote:

The more interesting question is who spends most energy per unit of mass.

That's actually a less interesting question imo.

From the linear equation approximation from the top google hit (caution, runners world):

Calories burned per mile = 0.72 x weight in lbs

For 200lb guy:

Total Cals for 13.1 miles = 0.72 x 200 x 13.1 = 1886 Cals

For 135lb guy

Total Cals for 13.1 miles = 0.72 x 135 x 13.1 = 1273 Cals

If you look at energy per unit mass then of course it simply cancels out and both of them are spending 0.72 Cals per mile per unit mass (or 9.4 Cals per unit mass for the 13.1 mile total).

Mr Elsewhere wrote:

E/kg wrote:

The more interesting question is who spends most energy per unit of mass.

That's actually a less interesting question imo.

From the linear equation approximation from the top google hit (caution, runners world):

Calories burned per mile = 0.72 x weight in lbs

http://www.runnersworld.com/peak-performance/running-v-walking-how-many-calories-will-you-burn

For 200lb guy:

Total Cals for 13.1 miles = 0.72 x 200 x 13.1 = 1886 Cals

For 135lb guy

Total Cals for 13.1 miles = 0.72 x 135 x 13.1 = 1273 Cals

If you look at energy per unit mass then of course it simply cancels out and both of them are spending 0.72 Cals per mile per unit mass (or 9.4 Cals per unit mass for the 13.1 mile total).

LOL. First order approximation in Runners' World.

Mr Elsewhere wrote:

E/kg wrote:

The more interesting question is who spends most energy per unit of mass.

That's actually a less interesting question imo.

From the linear equation approximation from the top google hit (caution, runners world):

Calories burned per mile = 0.72 x weight in lbs

http://www.runnersworld.com/peak-performance/running-v-walking-how-many-calories-will-you-burn

For 200lb guy:

Total Cals for 13.1 miles = 0.72 x 200 x 13.1 = 1886 Cals

For 135lb guy

Total Cals for 13.1 miles = 0.72 x 135 x 13.1 = 1273 Cals

If you look at energy per unit mass then of course it simply cancels out and both of them are spending 0.72 Cals per mile per unit mass (or 9.4 Cals per unit mass for the 13.1 mile total).

How can Time not be a variable in this equation? Each of the totals above should be multiplied by a factor based upon the time it took to complete the event.

Armchair Physicist wrote:

How can Time not be a variable in this equation? Each of the totals above should be multiplied by a factor based upon the time it took to complete the event.

Good question. Basically because it cancels out. Increased running speed means decreased total running time, and energy spent per mile has been measured as fairly constant across subjects of various size (I must be clear this is an approximation).

There is also a linear approximation for Calories burned per unit time in that article and unlike total energy spend, running speed affects Cals/min a great deal.

A dark horse arrives:

C: Runner 40 years old, 195 lbs @ 5'11", runs 1:26

Of course what we are really getting at is which is more impressive given some equivalency?

I was 155 pounds in the Army, so in theory I could get back there if I didn't drink soda and eat a bunch of junk all the time. That's 40 extra pounds. One method of compare is to see what the time "looks like" using various estimates.

Baseline pace is 6:35/mi.

At 1/s per mile, pace 5:55 for a 1:17:34

At 1.5/s per mile, pace 5:35 for a 1:13:11

At 2/s per mile, pace 5:15 for a 1:08:50

So I guess it depends what would happen if I was all of a sudden 155 pounds. I'd guess closer to the 1/s per mile considering diminishing returns, probably not a linear equation.

Conclusion, guy running 1:12 more impressive.

Mr Elsewhere wrote:

Armchair Physicist wrote:

How can Time not be a variable in this equation? Each of the totals above should be multiplied by a factor based upon the time it took to complete the event.

Good question. Basically because it cancels out. Increased running speed means decreased total running time, and energy spent per mile has been measured as fairly constant across subjects of various size (I must be clear this is an approximation).

There is also a linear approximation for Calories burned per unit time in that article and unlike total energy spend, running speed affects Cals/min a great deal.

I'd love to see the data. This "approximation" sounds like it's more a loose theory vs fact. If I slowly walk 2 miles in 25 minutes I'm certain I will use less calories than if I race 2 miles under 10 minutes.

In defense of the studies, they were probably completed on folks whose walking pace isn't much slower that their all out 2-3 mile pace.

Armchair Physicist wrote:

I'd love to see the data. This "approximation" sounds like it's more a loose theory vs fact. If I slowly walk 2 miles in 25 minutes I'm certain I will use less calories than if I race 2 miles under 10 minutes.

In defense of the studies, they were probably completed on folks whose walking pace isn't much slower that their all out 2-3 mile pace.

Walking 2 miles in 25 minutes is definitely NOT walking slowly.

Armchair Physicist wrote:

I'd love to see the data. This "approximation" sounds like it's more a loose theory vs fact. If I slowly walk 2 miles in 25 minutes I'm certain I will use less calories than if I race 2 miles under 10 minutes.

In defense of the studies, they were probably completed on folks whose walking pace isn't much slower that their all out 2-3 mile pace.

A "loose theory"? I do not know what that is.

It's a simple linear fit to energy expenditure measurements. All the links and sources are referenced from the RW article.

If I slowly walk 2 miles in 25 minutes I'm certain I will use less calories than if I race 2 miles under 10 minutes.

There are different linear fits for either running or walking. Again, all the details are in the article. Your statement above does not contradict anything.

So far, all the formulas presented overlook the vertical component of running. It's as if runners are being transported by frictionless wheels. In reality, runners oscillate up and down a couple of inches or so with each stride. (Some bounce more, some are less, but a couple of inches is a pretty good average guess.)

Surely that needs to be accounted for as well. If you run for an hour, about half that time is airborne, so the push off must overcome the force of gravity.

fisky wrote:

So far, all the formulas presented overlook the vertical component of running.

That's not really true.

A few people have mentioned: Work = force x displacement ( = mass x acceleration x displacement)

which is perfectly valid but simply difficult to solve for the system in question. However, the force/acceleration vector includes direction and hence does not overlook the vertical component.

The equations I have commented on are simple fits to empirical data, and as such implicitly include the vertical component of running given the data were gleaned from runners oscillating vertically.

My answer would only apply to spherical chickens in a vacuum.

thejeff wrote:Half Marathon

Recently had an unnecessarily deep conversation with my running buddy about impressive race efforts. Here is the gist:

Runner A: Speedy McLightweight: 135 lbs, finished in 1:12:00.

Runner B: Dee Lineman: 200 lbs, finished in 1:35:00.

So, who gave more effort?

I assume the question I need to ask is: Who did more WORK?

Work = Force x Distance, so W = F x (13.1mi, or 21. 08km)

So, what is Force? Mass times Acceleration.

So, Work = (M x a) x 13.1 miles.

This is where I get lost, if I am even barking up the right tree... how to quantify acceleration? Is it even necessary?

Is there an easier way to think about this?

2 pages of waffle & no answer !

the answers in undefined energy units is from :

work = 0.5*m*v^3

it is derived from 0.5*m*v^2 which is kinetic energy & another v factor from wind-drag at equilibrium pace

for skinny guy, his work is :

0.5 * 135 * ( 21097.5 / ( 72 * 60 ) )^3 = 7862 u

for fat boy, his work is :

0.5 * 200 * ( 21097.5 / ( 95 * 60 ) )^3 = 5071u

the skinny guy put in the most work

for fat boy to have put in same work, his time wouda been from :

7862 = 0.5 * 200 * ( 21097.5 / t ))^3

t = 1"22'04

It was definitely 1000 ducks that worked harder than the lion

ventolin^3 wrote:work = 0.5*m*v^3

From what did you pull this out of?

Mr Elsewhere wrote:There are different linear fits for either running or walking. Again, all the details are in the article.

This hitch here is that the kcal/mile/kg "constant" for walking was created by observing people walking only at 18:36 pace, and the running one was created by observing people running only at 10:00 pace. They didn't look to see what happens at different running paces (or different walking paces, or in the transition from walking to running) which to me is the interesting part.

Running vs. walking is a false binary, and I'd hesitate to assume that a kcal/mile/kg or kcal/min/kg estimate that works ok at 10:00 pace is still valid at, say, 6:00 pace. I have no evidence, but I suspect both rates will always increase with speed.

knox harrington wrote:

They didn't look to see what happens at different running paces (or different walking paces, or in the transition from walking to running) which to me is the interesting part.

That's true for one of the studies referenced. Other studies have. None have contradicted each other to my knowledge and I'll repeat 2 points pertinent to this discussion that have been demonstrated on numerous occasions.

1. Energy consumption per mile remains nearly constant across the range of normal running speeds (i.e. ~10min/mile and faster)

2. Energy consumption per minute (power/Wattage) increases with running speed.

Mr Elsewhere wrote:Other studies have.

If you have a link to one handy that doesn't require payment, I'd check it out. Otherwise I can google I guess.

Mr Elsewhere wrote:Energy consumption per minute (power/Wattage) increases with running speed.

Clearly, but the RW article you linked presents a kcal/min/kg figure.

Mr Elsewhere wrote:

fisky wrote:

So far, all the formulas presented overlook the vertical component of running.

That's not really true.

A few people have mentioned: Work = force x displacement ( = mass x acceleration x displacement)

which is perfectly valid but simply difficult to solve for the system in question. However, the force/acceleration vector includes direction and hence does not overlook the vertical component.

The equations I have commented on are simple fits to empirical data, and as such implicitly include the vertical component of running given the data were gleaned from runners oscillating vertically.

Thanks, but I'm still confused. Under the above formula, once a runner reaches a constant speed, no further work is done since horizontal acceleration is zero.

Example: If mass = 60kg and displacement = 1 mile, but acceleration = zero, then 60x1x0=0