Differential Equations Model

Question

Given the equation for sprinter velocity, dv/dt = M(u - v)/t

 

This is an ODE of 1st order and it's LINEAR, right?

Answer 1

Hi are you sure about the 1/t factor? That would make the acceleration infinite at t=0??  Not even Hussan Bolt is that fast. Otherwise it's a linear DE in v(t)

 

Jim

Answer 2

Jim, 

Thanks for following up... yours is exactly the kind of feedback I need. I recognize now that f(t) should actually be f (not a function of t), but I used JB Keller's 1973 formula for WR (World Record) sprinter acceleration and then integrated it with two constant terms, instead of f = 12.2 for acceleration, I updated that to 10.596 m/s² and I updated the physiological exhaustion rate from τ = 0.982 to τ = 1.274 (more modern data, revisions of Keller's work) and of course none of my calculations take into account wind resistance (-) or assistance (+) factors and no accommodations for a curved running track, the assignment does not require such considerations.

 

My overall project is to construct a modelling situation for sprinters speed in short duration races < 291m, since that is where the abundance of research lies about WR sprinting. So far, I've integrated the ODE and solved for C, then used  the initial value v(0) = 0 to get this formula:

 

v(t) = 13.499304(1 - e^-t/1.274)

 

Then I substituted numerous instantaneous times (1 sec, 5 sec, 10 sec, 1000 sec) to see if these values were reasonable, and sure enough, I came to the conclusion that the max running speed of a WR sprinter is about 29mph, even if that sprinter were miraculously able to maintain his (because this data is specific to male WR holders) acceleration beyond human capability - the max velocity with these constraints was 29.13... mph.

 

Thanks to your help, I'm a great deal closer to my final modelling project, thank you!