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.
 A: 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.
A: 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


*

*you accelerate to somewhat constant speed (actually you accelerate to an average speed but this should be a good enough approximation)

*You are advancing against air friction.

*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)
