# Running: Determine how much more energy is needed per extra kilogram of weight

(I recently asked this on maths but was directed here)

I have recently become a runner and having a keen interest in kinematics I'm very interested in the maths/physics of my running.

Can someone please help me determine a function that will enable me to determine how much harder I have to work to carry my 10kg overweight body over a given distance? I was pretty sure I could just use work=weight x distance for the flat roads, but someone has pointed out that my time would be meaningless in such a scenario, but I'm having difficultly determining this for the hills too.

In addition I need to include time as a variable so I can work out how my time affects my energy expenditure.

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I think this may be a biological question, as an animal will expend energy just standing still and holding a weight above the ground, whereas an inanimate object like a table does no work. From a physics point of view, you only do work against friction and the vertical line of gravity. And I can't see how to calculate friction for a runner. – metzgeer Jun 21 '11 at 1:21
Thank you. I'm looking to calculate an energy delta not the exact amount. Also lets assume friction is zero in the first case. I'm happy for a simple answer that assumes some ideals – Preet Sangha Jun 21 '11 at 1:38
Related questions: Convert running speed uphill to equivilent speed on flat, What's the difference between running up a hill and running up an inclined treadmill?. And let me add my endorsement to metzgeer's comment: this kind of problem is not well suited to a "pure" physics approach. – dmckee Jun 21 '11 at 3:52
I own a copy of some "Vademecum for engineers" from about 1820. There they make up the calculation for energy produced by marching soldiers as weight times distance ! Author is some Bernoulli, descendant of that famous family :=( – Georg Jun 21 '11 at 9:58

I have thought about this answer today while cycling. I have absolutely no idea how good is my solution'' but I am sure it isn't total b***it.

While running, you pay your energy mainly for three different reasons

1. you accelerate to somewhat constant speed (actually you accelerate to an average speed but this should be a good enough approximation)
2. You are advancing against air friction.

3. You are not running in a straight line but you are actually bouncing up an down.

In all three cases you are implementing force along a rout to generate energy (work) and you are transforming chemical energy (proton gradient, sugar, ATP etc.) into kinetic energy via muscle contraction.

If you are applying force to generate work, and the force scales like your mass then:

$F = ma \Rightarrow w = ma\Delta r$

Which means you more energy which is equal to your overweight times some constant. With good enough scales, you can estimate the excess energy which is consumed but running twice the same distance in as similar situation as possible (same time of day, same air conditioning etc.) once with a 10kg weight on your back and once with your clothing alone. weighting yourself before and after the run, mass difference can be translated to energy via some engineering tables (wikipedia states that one g of fat stores ~40kJ of energy)

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Excellent analysis. Thank you very much. – Preet Sangha Jul 8 '11 at 4:39
I think converting from a weight difference to calories used will be difficult, if its even possible to get a sensible answer. You also lose weight while running by expelling water as sweat and moist breath. – bdsl Nov 5 '15 at 23:31
This is why I said the conditions should be as similar to each other as possible, so we could assume that the vast majority of the difference step from the added effort. – Yotam Nov 6 '15 at 14:02

This really is a question about metabolism, you could refer to http://www.brianmac.co.uk/energyexp.htm and use the tables and calculator to find what your energy expenditure is.

Lets assume you are ten kilograms overweight and wish to find the energy expenditure (EE) for this excess mass. Find your EE for your excess weight say 110 kilos, next find your EE for your normal weight say 100 kilos, and substract the difference and that will give your EE for the excess 10 kilos.

To calculate the EE for going up a hill, find the difference in height:

Work = m g $\Delta$h

However the only real way of knowing your energy expenditure for such a messy physical system is to measure your metabolism, and I know this sounds silly but what if you counted the number of breaths you took as your ran? Your oxygen intake is directly proportional to your metabolism, that might give the best estimate of EE. Good luck.

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I know that is a messy system, with multiple variables. However you have pointed out the basic delta I need. With respect to breaths, I didn't know this correlation, and I will discuss it with my breathing physiotherapist. Once again thanks for an interesting off the wall anaswer. – Preet Sangha Jun 22 '11 at 8:57