The equations of motion of the runner: efficiency increases with increasing cadence

The story so far

In my post of 16th  Jan I presented results of a calculation of the work required between mid-stance and the achievement of peak height in the subsequent airborne phase to elevate the body from it’s low point at mid-stance, and to overcome the effect of braking in the first half of stance.  These calculations were based on a precise solution of the equations of motion for the centre of gravity (COG) of the runner’s body derived from Newton’s Laws of motion.  The comments by Ewen, Simon, Robert and Mike led to an extensive discussion of issues related to my calculations and the conclusions that I drew.   Here I will attempt a summary of the major issues that we discussed, including the evidence for validity of the calculations.

There are two assumptions implicit in my calculations.  First, that the time course of the vertical component of ground reaction force varies sinusoidally with time during stance (as shown in fig 1 of that post).  As can be seen by comparing figure 1 with force plate data (eg Figure 1c in the paper by Daniel Lieberman and colleagues, Nature, Vol 463,p 531, January 2010) this is a fairly good approximation to force plate data for a forefoot runner.    Secondly, that the tension in the leg muscles is adjusted to ensure that the total ground reaction force acts in the direction of the line form point support to the centre of gravity (COG).  This assumption is also supported by observational data, as outlined in the discussion between Simon and myself in the final few comments on that posting  (dated 31st Jan).

There were also two energy costs that I had ignored in my calculations: the effect of wind resistance and the effect energy consumed in re-positioning the limbs during the gait cycle.  In his comment on 22nd Jan, Robert kindly provided a fairly realistic estimate of the effect of wind resistance and demonstrated that for the examples that I was considering, the effect of wind resistance due to the runner’s own velocity through still air, as likely to be small.  I myself carried out some crude experiments to determine the energy cost of re-positioning the limbs, and demonstrated that these costs were fairly small and unlikely to alter the conclusions that I had drawn.  However the cost of repositioning limbs does increase with running velocity and the energy cost of repositioning the limbs is likely to be appreciable at higher running velocities. I will discuss this issue in more detail later in this posting.

Overall, the outcome of quite intensive discussion arising from comments by Robert, Simon and Mike is that the computation is likely to be a fairly good representation of the displacements, velocities and energy required to elevate the body and overcome braking in the case of a forefoot runner.  The main conclusion I had drawn was that the energy costs (for a given velocity and cadence) are lower when the time on stance is shorter.  In general a shorter time on stance can be achieved by maintain greater tension in the leg muscles so that the amount of flexion at hip and knee is reduced.  The intense discussion in the comment section helped consolidate my confidence that this is a valid conclusion.

It should be noted that my calculations of energy costs do not provide an estimate of the proportion of the required energy that can be recovered from elastic recoil.  However, because muscle and tendons are visco-elastic in the sense that their elasticity is greater when forces are applied my rapidly, it is likely that the amount of energy recovered via elastic recoil will be greater when time on stance is shorter, because the rate of increase in load is greater (as illustrated in fig 1 of my post on 16th  Jan)

The discussion included debate about two other less clear-cut issues.  First, I had initially argued that the greater forces and also the greater rate of increase in loading associated with a shorter time on stance presents a greater risk of injury.  I still believe that this is plausible, but both Simon and Robert pointed out that this is not necessarily the case.  It is necessary to compute the actual forces and shearing effects acting on particular joints and muscles to determine the risk of injury.  My calculations apply only to the motion of the COG and to the overall energy costs, but do not directly allow an estimate of forces acting at particular points in the body.  I think that my calculations might be informative regarding the stress on the legs, but further exploration of that issue is a topic for another day.  In particular, I think that the this topic has something useful to add to the debate about the merits of bare-foot running

The most hotly debated issue in the comments on my post was the implications of my calculations for rotational effects occurring during the gait cycle, and in particular for the controversial concept of gravitational torque.  In fact my model does provide a very clear answer regarding rotational effects.  There is indeed an increase in the angular momentum about a pivot point at the point of support in a head forward and down direction during the second half of stance, though the question of whether these should be described a consequence of gravitational torque is more debateable.  In the absence of wind resistance, this effect is cancelled by an opposite effect in the first half of stance, and I do not believe that this issue is of major importance in understanding running mechanics.  However, because it has been a bone of contention in relation to the theory underlying the Pose technique, I will devote a post to that topic in the near future.

Meanwhile, in this post I wish to deal with two issues.  First is the issue of cadence.  The second is a discussion of the energy costs of repositioning the limbs.


It is widely believed that increased cadence is associated with greater efficiency.  The energy cost (per mile or Km) of elevating the body to the peak height in the airborne phase decreases with increasing cadence.  This is because the duration of each gait cycle decreases as cadence increases.  The amount of gravitational potential energy lost when a body falls is proportional to the square of the duration of the fall.  Even though the number of steps per mile (or Km) increases linearly as cadence increases, because the energy saving is proportional to the change in the square of the duration, the energy saving more than offsets the increase cost due to an increase in number of steps.

As demonstrated in my post of 16th Jan, the equations of motion of the runner’s body provide a precise estimate of the elevation that occurs from mid-stance to peak height, and hence provide a precise estimate of the energy cost of elevating the body  These equations also provide a good estimate of the  energy required to overcome braking in early stance, provided data for hGRF is available as a result of either direct measurement or estimation based on vGRF (equation 5 on the calculations page).   In this post, I report the result of using these equations to examine the effect of increasing cadence, while maintaining a specified peak vGRF.

I have performed the calculation assuming a peak value of vGRF of 3g per Kg.  This value is midway between the two different values of peak vGRF I considered in my previous post, and is likely to be a fairly realistic estimate form many runners when running at 4 m/sec.   Figures 3 to 5 (numbered sequentially from the figures in my post of  16th Jan) illustrate  the braking effect and also the vertical displacement of the COG during the entire gait cycle, for the case where peak vGRF is 3g per Kg, for a cadence of 180 steps/sec and 200 steps/sec.  As expected, the vertical displacement is less at higher cadence.  There is a 19% reduction in the vertical displacement (and hence a 19% reduction in energy required to elevate the body in each stride), whereas the number of strides per mile (or Km) increases by only 11%.  Thus the energy consumption per mile (or Km) is about 8% less at 200 steps per minute compared with 180 steps per minute.

Fig. 4: Change in height of the COG from mid-airborne phase (in metres) when peak vGRF= 3*g Newton/Kg, for cadence 180 steps/minute (ochre) and 200 steps per minute (blue), at velocity 4 m/sec. Vertical blue lines indicate mid-airborne phase for cadence 200.

Fig. 5: Change horizontal velocity from mid-airborne phase (in metres/sec) due to braking when peak vGRF= 3*g Newton/Kg, for cadence 180 steps/minute (ochre) and 200 steps per minute (blue), at velocity 4 m/sec. Vertical blue lines indicate mid-airborne phase for cadence 200.

There is also a saving in the energy required to overcome the braking effect.  The duration of braking is shorter, due to the shorter overall gait cycle and associated shorter time on stance.  Furthermore, the leg is less oblique at footfall when cadence is greater and consequently the horizontal component of GRF is less.  As a result of both of these factors, the braking effect is less.   Table 2 gives the energy costs  of braking, elevation and total costs for a forefoot runner, running at 4 m/sec and peak vGRF = 3g per Kg for a cadence of 180 and 200 steps/min. The energy at the higher cadence saving amounts to approximately 14%.

Table 2: Mechanical work per Km per Kg required to overcome braking and to elevate the body, when peak vGRF=3*g per Kg, velocity = 4 m/sec.

A similar calculation performed for the situation where peak vGRF = 2g /Kg indicates reduction in energy cost from  1309 Nm/Km at cadence 180 steps per minute to 1186 Nm per Km at 200 steps per minute. Thus the saving is even greater when peak vGRF is lower (relatively longer time on stance) because the increased braking with greater obliquity of the leg at footfall is even greater at lower values of peak vGRF.

How do the estimated energy costs compare with directly measured metabolic costs?

It is of interest to compare these estimates of the costs of overcoming braking and elevating the body with evidence regarding the metabolic cost of running at 4 m/sec, though there are two uncertain quantities in determining the metabolic cost of achieving a specified amount of mechanical work to metabolic costs.  The first is the fact that muscle contraction is a relatively inefficient way of converting metabolic energy to mechanical energy.  It is generally accepted that during activities such as running and cycling, muscle contraction has an efficiency of approximately 20% (i.e the consumption of metabolic energy by muscle contraction is 5 times the amount of mechanical work done).   Secondly, when running a proportion of the energy required to overcome braking and elevate the body is derived via elastic recoil of muscles and tendons.  I am not aware of any data for the proportion of energy recovered by elastic recoil.  For present purposes, I assume that 50% of the energy can be obtained via elastic recoil.  Because the elasticity of tendons in greatest when that are loaded rapidly, this proportion is likely to be higher when the rate of loading of the muscles and tendons in early stance is highest.

According to the data published by McArdle in 2000 (Essentials of Exercise Physiology, USA: Lippincott Williams and Wilkins, 2nd ed. p170) the total metabolic energy cost of running at 4 m/sec is 62.2 Kcal/Km or 0.99 Kcal/Km per Kg (4142 Nm/Km per Kg). It is usually considered that this cost varies relatively little with variation in gait. While my calculation show that energy costs do vary appreciably with duration of time on stance, for the purpose of obtaining an approximate estimate of the relative proportion of total energy spent on elevating the body and compensating for braking, we only require an approximate estimate of total cost.  If we assume the efficiency of conversion of metabolic energy to mechanical work is 20%, while 50% of the energy can be recovered by elastic recoil, McArdle’s data indicates that the mechanical work done 1656 Nm/Km/Kg.  This is of course a crude estate and makes no allowance for the fact that time and stance and cadence produce modest but appreciable changes in energy cost.

Alternatively, using Daniels’ formula for oxygen consumption at paces in the aerobic zone, it can readily be shown that for a runner with VO2max of 51 ml/min/Kg, running at 4 m/sec (which is in the upper aerobic zone where energy is largely provided by metabolism of glucose (derived from glycogen) oxygen consumption is consumption 188 ml/Km per Kg.  When glucose is the fuel, 1 litre of oxygen provides 5.05 Kcal, giving a metabolic cost of 0.95 Kcal/Km per Kg, and corresponding mechanical energy cost of 1589 Nm /Km/Kg .  Thus Daniels’ data indicates a metabolic cost about 4% lower than that derived from McArdle’s data, but this difference is trivial in light of the uncertainties in estimating mechanical cost from metabolic cost.

The important conclusion is that whether one uses an estimate based on Daniels or McArdle, it is clear from the data shown in tables 1 and 2  (which indicate mechanical costs in the range 1180 Nm to 1400 Nm per Km per Kg)  that elevating the body and overcoming braking, make a major contribution to the energy cost of running,   Provided one starts with accurate data for GRF, it is possible to compute the mechanical work required to overcome braking and to elevate the body quite precisely, using Newton’s laws of motion (as indicated in my calculations page).   Assuming a sinusoidal time course of vGRF, the results are as presented in this posting and my post on  16th.   Nonetheless, the comparison with the data derived from McArdle or Daniels does indicate that there is some minor but nonetheless appreciable energy cost in addition to elevating the body and overcoming braking.

Repositioning the limbs

As mentioned above, the other appreciable energy cost of running is the energy required to re-position the limbs, especially the legs.  The foot must be accelerated from a stationary position on stance to a velocity somewhat greater than the velocity of the torso, so that it overtake the torso, by mid-swing, and then decelerates during late swing so that is near to zero relative to the ground at foot-fall.  In the discussion following my posting of 16 Jan, I described a crude estimate of the energy required to reposition the limbs based on estimate of the increased metabolic demand, based on measurement of increased heat rate, when I executed the arm and leg movements associated with repositioning the limbs,   The estimated cost of repositioning the limbs at pace of 4 m/sec and cadence 180 steps per minute was 0.28 Nm per step (208 Nm per Km) to achieve the range of motion required when maximum vGRF of 2g /Kg and 0.20Nm.Kg per step (or 150 Nm/Kg per Km) to achieve the range of motion required for a maximum  vGRF of 4g /Kg.  Thus , at a pace of 4 m/sec  the repositioning cost is only a minor fraction of total energy cost and, furthermore decreases as time on stance decreases, strengthening the conclusion that a short time on stance is more efficient.  The decrease in re-positioning cost when time in stance is shorter reflects a smaller range of motion and a longer airborne time in which to achieve repositioning thereby allowing a lesser acceleration.   I have not performed the corresponding measurements for cadence 180 and 200 steps per minute at maximum vGRF = 3g /Kg.  Although range of motion is less when cadence is higher, airborne time proportionately reduced demanding higher acceleration so it unlikely that the smaller range of motion required at higher cadence will offset the effect of a greater number of steps per Km.  I will perform further measurements of the costs of limb repositioning in the near future.  Whatever the outcome of these measurements, the fact that repositioning costs are only a minor proportion of the total at a speed of 4 m/sec makes it unlikely that further measurements will appreciably alter the strength of the conclusion that it is more effect to run at a higher cadence.

In an article in the Journal of Physiology in 1977 (volume 268, p467), Cavagna and Kaneko estimated the energy required to reposition the limbs based on measurement of limb movement derived from video recordings of runners. They conclude that it exceeds the energy associated with overcoming external forces at speeds above 20 Km/hr (5.55 m/sec). However, they acknowledge that there are many uncertainties in their calculations. Nonetheless when estimating energy costs at high speed it is likely to be important to take account of re-positioning of the limbs.   Actions such as flexing the knee of the swinging leg so that the lever arm is short would be expected to have an appreciable effect at high speed, but probably matters little at speeds of 4 m/sec or lower.

What determines the upper limit of cadence?

While, I am confident that the mechanical work required to overcome braking and elevate the body decreases as cadence increases, this does not prove that metabolic efficiency will continue to increase with increasing cadence.  In estimating the relationship between mechanical work and metabolic cost of running, we had to take account of two variables: the efficiency with which muscles convert metabolic energy to mechanical work, which is typically about 20%, and the proportion of energy that can be recovered via elastic recoil.  As cadence increases, there will come a point at which the contraction becomes less efficient, and in addition, it might also be expected that recovery of energy via elastic recoil will also diminish.  Hence there will be an upper limit to the optimum cadence.  I suspect that the upper limit will be determined by the peak rate at which muscle can generate the tension required to capture kinetic energy and convert it to elastic energy.   The observation that elite  5Km and 10Km runners tend to exhibit a cadence in the range 180 to 200 steps per minutes suggest that the peak is around 200 steps per minute.

Interim conclusions and issues for future analysis

Overall, my calculations indicate that efficiency is greatest when time on stance is short and cadence high.  It is current folk lore among runners that you should land under the COG.  That is impossible, when running at constant speed except when running into a strong wind, because the push from hGRF when the point of support is behind the COG would inevitably cause continued acceleration.   However the twin principles of short time on stance and high cadence are the principles that allow the runner to minimise the amount the foot is ahead of the COG at footfall.   These calculations simply use the principles of physics to explain why Tirunesh Dibaba runs like this.

In the near future I will address three further issues:

1)      How much does a change in the time course of vGRF from that typical of a forefoot runner to that typical of a heel-striker affect the energy costs.

2)      Does the increase in angular momentum around the pivot at the point of support, due to the effect of gravity, play an appreciable role in the presence of wind resistance.

3)      What are the implications of these calculation for risk of injury and in particular, for the potential benefits or costs of barefoot running?

9 Responses to “The equations of motion of the runner: efficiency increases with increasing cadence”

  1. Ewen Says:

    Canute, briefly going back to the previous post and your comment about using light ‘Tiger’ shoes to run marathons I’d agree that one can learn to run in minimal shoes but running barefoot isn’t the way to go for racing on the road.

    With the energy cost of repositioning the limbs, wouldn’t this come down as speed increases? For example, if one were to run slowly, but replicating the limb movement of Dibaba there’s going to be a high energy cost as one pulls the foot up near the buttocks. If one is running fast or sprinting there’s minimal energy cost as the movement is just a reaction to the foot coming off the ground.

    Interesting figures about energy cost coming down as cadence increases (up to a point). A way of testing this might be to run on a treadmill moving at a fixed speed and use different cadences/stride lengths and see how HR varies. There’s probably going to be an ‘ideal’ cadence for each person when running at threshold speeds. Two runners are going to have different ‘amounts’ of elastic recoil for instance.

    • canute1 Says:

      Thanks for your comment. You raise a good point about the role of recoil in reducing the energy cost of repositioning the limbs, at higher velocities. The paper by Cavagna and Kaneko (Journal of Physiology in 1977 (vol 268, p467), that reported increasing repositioning with increasing running speed was based on measurement of the speed of movement of the various limb segments and calculating the work required to produce the observed movements. However, this approach does not tell us where the energy comes from. In particular it is possible that elastic recoil might make a substantial contribution My own estimation of the repositioning costs was based on an estimate of metabolic energy required and hence provides an estate of the cost after discounting energy provided by recoil. However I compared the different ranges of motion associated differing duration of stance at constant speed. My evidence merely indicated that at modest speed (4 m/sec) repositioning costs appeared to be only a minor fraction of total energy cost, but did not directly address the question of what happens at higher speed.
      Nonetheless, I think that it is likely that repositioning costs do increase with speed. You refer to lifting the foot towards the buttocks, but I think that at high speed, the big cost will be the horizontal acceleration. When an object is accelerated to reach a target speed within a fixed time limit, the energy required is proportional to the square of the acceleration. The swing leg, which must accelerate from rest to a speed somewhat greater than the speed of the torso between lift-off and mid-swing, and hence as running speed increases, must accelerate increasingly rapidly.
      However, as in the case of the calculations of the calculations by Cavangna & Kaneko, this argument does not rule out the possibility that recoil might contribute a greater proportion of this energy at higher speed. Some coaches (eg Steve Magness: emphasise the importance of hip extension at the end of stance to pre-load the hip flexors. This preload will store elastic energy that helps drive the forward swing. It is noticeable that in elite 5000m and 1000m runners (including Dibaba) the toes remain on the ground well after the heel has lifted and most of the unweighting has occurred. This apparent ‘delay’ on stance in the final moments before lift-off might add a little extra preload of the hip flexors. However, in view of the loss of efficiency with prolonged time on stance (illustrated in my post of 16th Jan) I do not think it is desirable to deliberately aim the keep the foot on the ground. My own experience is that it is best to focus consciously on a swift lift off. However I cannot rule out the possibility that even when I focus on swift lift-off there is appreciable preloading of hip flexors.
      With regard to your suggestion that one might determine one’s optimum cadence by measuring HR while running on a treadmill at various cadences, I think in principle that is worth trying. I do not have easy access to t treadmill, though I have done a similar thing on the elliptical, and have demonstrated that I am most efficient at a step rate in the range 190-195 / min. When running (outdoors) at tempo pace I appear to be most efficient at around 195-200 /min, but on account of the various factors that affect outdoor running, the measurements are less reliable. Nonetheless with a Polar foot pod, it is posible to get failry consistent measurement of cadence

  2. Ewen Says:

    Thanks Canute. I recall those blogs from Steve Magness about ‘how to run’ and hip extension… that the power comes from the hips and that the knee coming forward isn’t a conscious action.

    I had easy access to treadmills when I was in the States — wish I’d thought to try different cadences. Something to do next time. When I’ve counted it, my ‘natural’ cadence at a steady pace is in the low to mid 170s. 180 took a conscious effort and anything higher (if I wasn’t sprinting) seemed unmaintainable.

  3. Simon Says:


    In your paragraphs on limb repositioning, you mention an increase in air time when time on stance is shorter. That is not necessarily the case. An increase in cadence and a decrease in stance time can lead to no change in air time.
    I think it would be best to consider stance time, cadence and limb repositioning together as it is an inter-related system.

    Limb repositioning costs decrease when; stance time is shorter (less limb displacement), pace is slower (same), cadence is slower (limb repositioning is less rapid) and air time is longer (more time to reposition limbs).

    I think a better estimate of limb repositioning is necessary – I seem to remember there are some calculations in the public domain that have projected costs? I seem to remember that the cost of repositioning limbs became dominant at sprint speeds when compared to the cost of getting airborne. I also remember it being significant at more moderate paces.

    Whilst you may want to ignore limb repositioning costs at slow paces, I don’t think that can be reasonably done as the cost does not only rely on pace; a runner could run a very slow pace at 200 steps per minute and it wouldn’t surprise me if repositioning was the largest energy cost in that scenario.

    Another small point is regarding the upper limit of cadence. You say it should be what is observed in elite 5k runners – I would suggest that is more likely to be a near optimal cadence for their pace and naturally selected limb recovery rate rather than an upper metabolic limit.

    I hope you take my comments constructively as they are meant, I am impressed with what you are doing here and think it a very worthwhile project.

    • canute1 Says:

      Thank you for your comment. I agree that we are dealing with a complex system in which many different factors must be taken into account. In particular, I agree that limb repositioning costs must be considered carefully.

      My own thinking about the subject, supported by your helpful list of the factors that determine limb positioning costs, leads me to believe that in the range of cadence values, stance times and speed that I have considered in this post and my previous post on Jan 16th, the decrease in repositioning cost associated with decreased time on stance considered on 16th Jan will be greater than the increase in repositioning costs due to increase cadence from 180 to 200 steps per minute. I believe this is the case partly because the proportionate decrease in displacement associated with the change in time on stance is greater than the proportionate change in time available for repositioning due to higher cadence. Granted that the available evidence suggests that repositioning costs are less than the combined cost of braking and elevating the body at 4 m/sec (based both on the data presented by Cavagna and Kaneko (Journal of Physiology in 1977 (vol 268, p467), and on my own crude estimate of repositioning costs for the circumstances presented on Jan 16th) I am still fairly confident that repositioning costs will not change the main conclusion that I draw in this post. However, as has become clear in the discussion between yourself, Klas and myself in response to my post on Jan 16th, these conclusions might not apply under other circumstances. In particular, I suspect that they might not apply at speeds much less than 4 m/sec (eg at 2.5 m/sec, as discussed in response to Klas) and speeds much greater than 4 m/sec (eg 5.5 m/sec, as discussed by Cavagna and Kaneko).

      With regard to the upper limit of cadence, I based this on both elite 5K runners and on observations of myself – certainly very far from elite in my present state. When sprinting I appear most efficient at a little above 200 steps per minute, and at fairly slow paces (eg 3 m/sec) I appear most efficient at around 190 steps per minute.

  4. Maria C Says:

    Hi I am doing a physics project on the efficiency of different running cadences. Could you help me out with how to calculate the energy of a person at different cadences so I can then calculate the efficiency. Thanks,


    • canute1 Says:


      The main mechanical costs are the cost of getting airborne, the cost of overcoming braking when the foot is one the ground, and the cost of moving the swing leg forwards during the swing phase. There calculations are very complex. In my calculations pages I have given formula for calculating each of these (based on certain simplifying assumptions). However, computing the answer from these equations requires a computer program which performs numerical integration.

      There are also internal costs due to inefficiency of conversion of metabolic energy to mechanical energy in muscle. These cannot be computed, but can be estimated form knowledge of the amount of oxygen consumed and measurement of a quantity known as the respiratory change ratio which indicates the proportion of glucose to fat used as fuel.

      So I am afraid the computations are very complex. Using the equations given in my calculation page, for a person with centre of mass at a height of 1 metre, running at a speed of 4 metre/sec, the mechanical energy costs (in Newton – metre per Kg of body weight) are:

      cadence 180 steps/min
      work overcoming braking 586.1 N-m
      work elevating the body 616.6 N-m
      work repositioning the legs 660 N-m

      cadence 200 steps/min
      work overcoming braking 533.5.1 N-m
      work elevating the body 557.7 N-m
      work repositioning the legs 733.3 N-m

      These numbers indicate that the total mechanical cost at 200 steps per minute is a little less than at 180 stesp/min. However, these numbers do not include the internal cost of conversion of metabolic energy to mechanical energy. Therefore the results are apporximate and you can only conclude that costs are very similar at 180 steps/min and 200 steps/min at a speed of 4 m/sec. At much slower speeds,,lower cadence is more efficient..

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