280 BIOPHYSICAL STUDIES ON NERVE AND MUSCLE 



that momentum, mv, is conserved (remains constant): 



£) I — 1 = constant 



\ max / \ max / 



where v max is the maximum speed of shortening (no load), and ra max is the 

 largest weight which can be lifted. Here v and m are constants. This result 

 is often written as a product of velocity and force, when the acceleration is 

 that due to gravity — a constant; it then becomes the "force- velocity rela- 

 tionship." Thus for two masses, m, and m 2 , momentum conservation is 

 expressed 



<m x — m \ lv~, — vAfm 2 — m^ 



max / \ max / \ max / \ max 



The denominators cancel out. Then if mass 2 is chosen to be just big enough 

 that the muscle can sustain it but not lift it, v 2 = 0, and m 2 = m max . Mul- 

 tiplication through by g, the acceleration due to gravity, converts masses to 

 forces (F = mg), and then rearrangement gives 



(F+a)(v + b) = (F. + aji, 



the force-velocity relationship, first stated in 1938 by A. V. Hill (a and b are 

 his constants, equal to -gm and -v respectively). Figure 10-10 illustrates 

 this equation, and says simply that the greater the force to be overcome by 

 the contracting muscle the less the speed at which it can contract. Rear- 

 ranged in the form 



v = (F max -nb/(F+a) 



it says that the velocity of shortening depends upon the difference between 

 the maximum force it can develop and the actual force on the muscle. This 

 hyperbolic relationship is obeyed by a wide variety of muscle and muscle 

 systems, including the human arm. 16 



On Energetics of Muscle 



The work done by the muscle in lifting a weight is given by the product 

 mgh, where g is the acceleration due to gravity (and therefore mg is force, since 

 F = ma) and h is height to which the weight is lifted. We saw in Chapter 7 

 that part (ALT') of hydrolysis of ATP — a reaction catalyzed by the con- 

 tractile enzyme, myosin — could appear as work of contraction. Thus: 



A5' = mgh 



and the rest of the total free energy of reaction (A?F) is wasted because of ir- 

 reversibility or inefficiency in the process, and thrown away as heat, TAS '. 



