Bodies in Motion: Human Limitations

Muscle Force, its relation to Speed and Power delivery.

The muscle can deliver the largest force at zero velocity of the muscle contraction. The muscle can contract with a certain muscle speed which is dependent of the length of the muscle. Typically, the muscle can contract at between 2 and 4 times the length of the muscle per second. Both of these emperical observations results from the microscopic muscle contraction process.

There are molecular motor in the muscle which draw micro sections of the muscle together. These motors have a intrinisc maximum speed. This speed is approximately: . The higher speeds observed for macroscopic muscles [groups of these fibers] is the result of the end-to end joing of the fibers producing N times the speed of contraction per fiber where N is the number of fibers joined together end-to-end.

The molecular motors must have traction to draw the muscle substructures together for a contraction. The efficiency of this traciton is highest for zero or low speeds but deteorates with higher speed. This slippage of the molecular motors results in lower forces as the muscle contraction speed is increased.



Recall that the power produced by a muscle is P = F * velocity. At zero velocity, no power is produced and at the highest velocity, F=0 and no power is produced. For the functional shape of the force-speed curve, the maximum power should occur at about 1/3 maximum speed. This is born out by the maximum power production of cyclists at about 100 rpm. Cyclists, in training, can obtain 300 rpm with no load which makes the maximum power output at about 1/3 max speed. A cyclist can continually adjust the gears on the bike so that he can be pedalling at the optimum muscle speed independent of the conditions [hill, wind].

Figure showing the force/power vs muscle speed. The ordinate is in units of maximum force and the abcissa is in units of the maximum speed for the muscle. The secondary ordinate is in units of power scaled by maximum force and maximum velocity. Peak power occurs at about 1/3 maximum velocity with a power of 1/10. The muscle can develop the highest forces when it is being stretched.[negative velocity].


In everyday language, we sometimes use power and energy interchangeably. Here we will be quite strict about proper useage. Power is the time rate of work.

Power = work/time

watt = joule/sec

If the athlete lifts the 100 kg barbell by 2 m and this is done in one second, the power is P = work/time = 2000 J/1 sec = 2000 watts.

Analyzing the Jump Reach

The Jump Reach is a standard test used by pro scouts to evaluate athletes. Basically the Jump Reach is a stationary vertical jump in which the athlete tries to touch a scale on the wall as high as possible. What is measured is the height of the jump, but what is inferred is the power output. What follows is an analysis of a Jump Reach by a UW football player.

The athlete remained in the air for 0.76 seconds which implies a jump reach height of 0.72 m. The time from the swing of the arms until the feet leave the floor is a total of 0.33 seconds. The final thrust of the legs takes about 0.13 seconds and the leg thrust raises the body from the squat position to a standing position, a height of about 0.3 m. The complete jump then has raised the CM of the athlete by a height of 1.02 m. The mass of the athlete is about 100 kg. If most of the energy comes from the leg thrust, the power that must be developed is:

Average Power = Energy/(time of the leg thrust) = mgh/(time of the leg thrust)

P = 100*9.8 * 1.02/0.13 = 7700 watts. This is a lot compared with the power exhibited during the stair climb of the class members (which was about 800 watts on the average). As the athlete moves faster near the end of the jump, the power requirements are even higher. Max Power = 2*(average power) = 15400 watts!! [assuming a constant acceleration]

If we average the power over the complete cycle of arm swing and torso thrust, the power required is much less.

Max Power = 2*100*9.8*1.02/0.33 = 6060 watts which is still quite high.

The important power effects are probably in the 0.2 second interval when the body and arms begin their upward journey. This makes the power average to be 5000 watts with a maximum power required of about 10,000 watts which is a lot! Some of the 140 kg linemen can jump this high so that they produce power proportionately higher.

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