This problem was contributed by Reed Gurchiek at Clemson University.

Furusawa et al. (1927) proposed a simple model for sprint running. The sprinter is modeled as a point mass (

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) with one degree of freedom (straight-line translation in the direction of the sprint). The model predicts forward velocity as a function of time (

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) and two constants (

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and

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):

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(a) Let

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be the force responsible for accelerating the sprinter forward. Show that the force

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predicted by this model is linear in the sprint velocity

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. Write the linear equation in slope–intercept form. (Hint: The slope and intercept are functions of the constants

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,

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, and

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.)

(b) At what velocity does the linear equation from part (a) predict that the force

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is 0? According to the model, at what time

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would the sprinter reach this velocity?

(c) Let

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be the rate at which

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does work on the sprinter (i.e.,

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is the power delivered by

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). Recall that the power delivered by a force on a point mass is equal to the product of force and velocity. Show that

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is quadratic in the sprinter's velocity

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.

(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

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and the intercept of the linear equation you derived in part (a) (use

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to denote the intercept).

(e) Let

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and

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so that the above equation approximates the velocity–time relationship of a runner during a particular sprint. Suppose the sprinter starts running at

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with an initial position of

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. At what time will they reach

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? (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.

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