Hill’s law: When a force is applied to muscle tissue, the muscle contracts. Hill’s law is an equation that relates speed of muscle contraction with force applied to the muscle.73 The equation is given by the rational function
where S is the speed at which the muscle contracts, F7 is the maximum force of the muscle at the given length 7, F is the force against which the muscle is contracting, and a and b are constants that depend on the muscle tissue itself. This is valid for non-negative F no larger than F7. For a fast-twitch vertebrate muscle—for example, the leg muscle of a sprinter—we may take F7 = 300 kPa, a = 81, and b = 6.75. These are the values we use in this exercise.
a. Write the equation for Hill’s law using the numbers above for fast-twitch vertebrate muscles.
b. Graph S versus F for forces up to 300 kPa.
c. Describe how the muscle’s contraction speed changes as the force applied increases.
d. When is S equal to zero? What does this mean in terms of the muscle?
e. Does the rational function for S have a horizontal asymptote? What meaning, if any, does the asymptote have in terms of the muscle?
f. Does the rational function for S have a vertical asymptote? What meaning, if any, does the asymptote have in terms of the muscle?