Muscle Contractions


The total force F exerted by the muscle is proportional to the number of contracting fibers and thus to the cross-sectional area of the entire muscle.  This implies that the maximum contraction speed V of a muscle is independent of the size of the muscle.  The speed V is therefore said to be scale invariant.  When a muscle contracts while exerting a force f > 0 (such as while lifting a weight of magnitude f), it’s maximum contraction speed v is less than the maximum contraction speed V obtainable when the muscle exerts zero force (f = 0).  This maximum speed v decreases when f increases, and becomes 0 when the muscle exerts it’s maximum force f = F.  The power p = fv exerted by the muscle is therefore 0 when f = 0 and v = V, and also when v = 0 and f = F.   

A consequence of scale invariance is that all animals of similar shape (racing dogs and horses, cats and tigers, dolphins and whales) run (swim) at approximately the same maximum speeds.  Running is, however, a relatively inefficient process because, after a runner’s rear leg propels his body off the ground and the forward moving front leg strikes the ground, the front leg exerts a backward force on the body, which causes the body to temporarily slow down.  This breaking effect is obviously inefficient, but elastic energy transfer to and from the Achilles tendon greatly increases the efficiency of running.

Walking, unlike running,  is not scale invariant. The most efficient walking speed occurs when the moving leg swings forward as a pendulum, powered mainly by gravity, with minimal muscular contribution. This speed is proportional to the square root of the leg length of the walker.  Maximum walking speed is also proportional to the square root of the leg length of the walker.

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  Power Consumption


The energy source for muscle contraction in not aerobic but is the chemical conversion of ATP into ADP within the muscle.  The role of O2 (inhaled into lungs and transported to muscles by blood circulation) is to covert the ADP back into ATP. The maximum aerobic power (MAP) of a person is his maximum rate of oxygen consumption obtained from the air inhaled into his lungs.  The greater a person’s MAP, the greater will be his ability in endurance activities such as long-distance running.  In addition, each person has an anaerobic store of energy available for muscle contraction. It is well-known that walking is more efficient at slower speeds and running is more efficient at higher speeds.  People change gaits from walking to running when it is energetically favorable, i.e., when it requires less power to run than to walk.

A person riding a bicycle can move much faster than a runner, with much less power consumption.  There are two reasons for this. 

  • (1) There is no backward-pushing breaking phase in cycling.  As a cyclist’s leg rotates a bicycle’s pedal, the applied force of the bicycle on the ground is always in the forward direction. 
  • (2) The many available gears on a bicycle enable the cyclist to choose a gear that enables him to exert the optimal force and speed that minimize his required power consumption.

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  Optimal Velocity in a Race


We have used the biomechanics of muscle contraction and oxygen consumption described above, together with the laws of mechanics and thermodynamics, to determine the optimal strategy for a runner to use in a racing competition.

Let v(t) be the (positive) speed of the runner at time t during a race of distance D, let f(t) be the forward force exerted on the ground by the runner at time t, and let E(t) be the internal energy available for running at time t.  The problem to be solved is to determine how the runner should choose f(t) in order to minimize the time T required to run the distance D, subject to the constraints f(t) ≤ F and E(t) ≥0, for all times 0 ≤ t ≤ T, where F is the runner’s maximum possible exerted force.  The functions f, v, and E are related by Newton’s second law of motion and the first law of thermodynamics.  Our solution to this problem, obtained by applying optimal control theory, is the following:  f(t) = F  from t = 0 until a later time t = t1, v(t) is constant between t = t1 and a later time t = t2, and E(t) = 0 from t = t2 until t = T.  The values of t1 and t2 depend on the value of D.  If D is small enough, t1 = t2 = T, and the solution is f(t) = F throughout the race. These races are called sprints. For longer races, the values of t1 and t2 are less than T, and are determined by the optimization equations.  For these long-distance races, the fact that the theory predicts that v(t) is constant throughout most of the race is in agreement with what is observed in long-distance races.

For D less then about 300 m, the races are sprints, and the average speed increases with increasing D because the speed in a sprint steadily increases with increasing t.  For the long-distance races, the average speed decreases with increasing D because slower speeds require smaller energy consumptions.

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