Imagine using your muscle power to lift a mass. If the mass is small, you can lift it quickly, as the force exerted on the mass by gravity is small. If the mass is larger, you would only be able to lift it more slowly, while if the mass was very large you might not be able to lift it at all, or it might just pull your arm down (instead of your arm moving the weight up).

Clearly, there is a relationship between the size of the mass and the speed at which you can lift it. This relationship is the Hill force-velocity equation, which says that the speed, v, at which a muscle contracts against a force, p (which is usually called the load), is described by the equation

(p + a)v = b(p0 − p), where

a, b and p0 are constants determined from experimental This equation was first derived by A.V. Hill in 1938, the same physiologist who is famous for the Hill equation that we discuss in a number of places in this book (see, for example, page 265 and Fig. 12.9).
data. The data from Hill’s original 1938 paper are shown in Fig.
12.14, together with the fit to Hill’s force-velocity equation.

a. Write v as a function of p. What are the asymptotes of the force-velocity equation, v(p)?

b. What is the scientific interpretation of the constant p0?
(Hint: what is the maximum load against which a muscle can contract?)

c. How would you interpret the value v(0)?

d. Does it make sense for v(p) to be negative when p gets large?

e. Does it make scientific sense for there to be a horizontal asymptote?