A calculation gone wrong; calling all physicists for help

3rm and I have now been diligently doing our morning running ritual for about 7 weeks (not all consecutively, given my propensity to fall ill from time to time) and it has reached that point where arguments ensue (which for me and him is an inevitability) about how best to improve.

His desire: to tone up. Mine: to get fit. His goal: to sprint 4.6 bloody kilometres (half of that is uphill). Mine: to run 8-10 km without feeling dead afterwards.

So, this research was all prompted by a huge (breathless) screaming match we had while lugging our exhausted bodies back up Oxford Rd, much to the bewilderment/amusement of commuters waiting for their public transport at the robot.

The argument

3rm: to tone your muscles, you’ve got to damage them with harsh exercise, causing them to build up scar tissue which is more solid than regular tissue. So we need to speed up our run radically, [insert unnecessary curse words here].

Ant: eeuw. Why? To get fit, you just need to run long distances, it doesn’t matter whether you speed up drastically, you just need to keep the heart beating at an elevated level for a reasonable period of time, [retort with equally vile battery of cussing].

3rm: No! We must go faster and do anaerobic exercise, $%*^(&$!

Ant: You can’t keep up anaerobic exercise levels that long buddy, trust me. Let’s do it the way nature intended, ok? There’s a reason that complex life evolved on this planet, and one of the main reasons for that is oxygen, you [bleep bleep bleeeeeeeeeeeeeep].

Of course, the answer is that neither of us is right nor wrong – the right type of exercise depends on what we want to achieve by exercising in the first place. I do like to think that I’m more right [naturally] but I can even justify my claim: aerobic (i.e. oxygen-burning) exercise is best suited for low-intensity exercise over a longer period of time, and I don’t think that a 4.6 km run can ever be short enough for people of our fitness level to be realistically maintained anaerobically (i.e. without burning oxygen).

The two types of energy consumption are also interchangeable, so that during a relatively fast-paced run, you’ll be using the lactic acid (anaerobic metabolism) cycle to some extent, with the aerobic cycle dominating (and vice versa for high-intensity burst of activity, e.g. a sprint, weight lifting, jumping). The trick is to find the correct exercise intensity level at which any lactate produced from the lactic acid cycle is rapidly and thoroughly consumed by your body – you don’t want this accumulating in the muscles as it leads to cramp, and thus has a detrimental effect on muscle function. The good news is that the more training you do, the greater this so-called lactate threshold (or anaerobic threshold) becomes – i.e. you will be able to train at higher exercise intensity levels before lactate builds up in your body, causing cramps. 3rm and I have certainly seen evidence of this, as we are now able to run the whole uphill portion without stopping to catch our breath anymore. I just don’t see the need to escalate our pleasant run to a sprint – I’d rather run further.

Out of interest, I also wanted to see what causes the greater consumption of energy (for a weight-loss perspective, a measure neither of us is using) – an increase in speed or distance. So I’m going to do the dangerous thing and put some equations up for all the geeky (and more scientifically adept) people to scrutinize. Warning: my physics is somewhat lacking (so if I’m using the wrong equations, please speak up and enlighten me):

For the effect of speed:

E = 1/2 mv^2 (energy consumed = 1/2 x mass of the body x velocity squared)

For a 10% increase in velocity:

E = 1/2 m (1.10v)^2

i.e. E = 1/2 mv^2 x (1.21) (taking the factor of 1.21 out to the end)

i.e. a 10% increase in velocity results in a 1.21 times greater energy consumption, which for those of you have forgotten all your high-school maths, is a 21% increase in energy consumption.

For the effect of distance:

W = Fs (Work done, which can be equated to energy transferred, = force x distance run)

What I’m going to be ignoring in my calculations is the effect of friction, which adds a significant amount to the extra work that needs to be done to keep moving)

For a 10% increase in distance:

W = F . (1.10s) (assuming your force, dependent on your mass and acceleration, is constant)

Therefore W = (Fs). (1.10)

i.e. a 10% increase in distance equates to a 10% increase in work done (or energy consumed, but like I said, this excludes the work done to overcome friction)

I guess these results make sense if you look at the equations – the speed equation is a power relationship between work and speed/velocity, whereas it’s a linear relationship for work and distance. You can even investigate this the lazy way – when next you’re in the gym, memorise your kJ consumption rate at each speed you run at, plot it on a graph and voila! You’ve plotted your first parabola since Matric.

Of course, I’ve just spotted a major omission in the velocity calculation – there’s no accounting for the length of time for which this velocity is kept up, which will obviously impact on the total energy consumed. I’m guessing you’d have to plot v^2 against time, and the area underneath it (multiplied by half your mass) would be equal to the total energy consumed – but this means the energy is a factor of 2 variables now, and… I give up. I hope one of you can help me.

So, it seems I’ve strayed from the original intent of this post – to disprove 3rm. I guess we’re both right, and if it is a weight-loss objective you have in mind, I’ve unsuccessfully sort-of proved that speeding up is more effective than running further, given a similar percentage improvement in either.

Oh bother. Being a bad scientist sucks.