3.8 Furusawa's sprint model

Furusawa et al. (1927) proposed a simple model for sprint running. The sprinter is modeled as a point mass () with one degree of freedom (straight-line translation in the direction of the sprint). The model predicts forward velocity as a function of time () and two constants ( and ):

(a) Let  be the force responsible for accelerating the sprinter forward. Show that the force  predicted by this model is linear in the sprint velocity . Write the linear equation in slope–intercept form. (Hint: The slope and intercept are functions of the constants , , and .)

(b) At what velocity does the linear equation from part (a) predict that the force  is 0? According to the model, at what time  would the sprinter reach this velocity?

(c) Let  be the rate at which  does work on the sprinter (i.e.,  is the power delivered by ). Recall that the power delivered by a force on a point mass is equal to the product of force and velocity. Show that  is quadratic in the sprinter's velocity .

(d) Using the quadratic equation you found in part (c), calculate the maximum power output and the velocity at which it occurs. Write your answer in terms of  and the intercept of the linear equation you derived in part (a) (use  to denote the intercept).

(e) Let  and  so that the above equation approximates the velocity–time relationship of a runner during a particular sprint. Suppose the sprinter starts running at  with an initial position of . At what time will they reach ? (Hint: You will need to use a nonlinear solver like the fsolve function in Matlab.)

Reference
Furusawa, K., Hill, A.V., Parkinson, J.L., The dynamics of "sprint" running. Proceedings of the Royal Society B 102: 29–42, 1927.