Question

Consider a free-falling bungee jumper whose vertical velocity is given by
(1) dv/dt = g - (cd/m)v2
The drag coefficient cd is actually not constant but increases as the bungee jumper falls because air is more dense lower in the atmosphere. We suppose that
(2) dcd /dz = cd/H
where H is a scale height of the atmosphere. (Note that v = dz/dt.) Suppose that g = 10 m s-2, m = 60 kg, and H = 8 × 103 m. The initial conditions at t = 0 s are v = 0 m s-1 and cd = 0.2 kg m-1. We want to estimate v at some later time(s).

(a) Show how you would apply Euler's method to solve this problem

Answer 1

The professor said we should write an approximate expression for v(t+h) in terms of values of variables at time t

Answer 2

euler's method can be utilized if you have a complex root for your characteristic differential equation. I suggest that you bring all terms to one side and set them equal to so. Because dv/dt is acceleration you will the second derivative of your position function and that will give you a linear second order differential equation. Then you can substitute various constants for parameters such as cd to make the calculations simpler. I will work on it myself and try to post a solution later

Answer 3

Thank u so much, I will try that

Answer 4

I'm not sure if it'll work but you're welcome!

Answer 5

Also you have to be aware that for later time(s) the vertical velocity will be approximately closer to g and that will become the terminal velocity. You might get a result such that as your v(t) decays exponentially it never approaches 0 but there is an upper limit for the position function of the bungee jumper and that upper limit is what we call terminal velocity.

Answer 6

So i know that the analytical solution would be v(t) = sqrt(gm/cd) tanh(sqrt(gcd/m)t)
And the Euler method would be Vi+1=Vi + Fi (Tn+1 - Tn) then for
dv/dt = g - (cd/m)v2 we would have something like
Vi+1=Vi + h(g - (cd/m)v^2) where initial condition Vi(0)=0(given)
but for dcd /dz = cd/H since v=dz/dt can we assume that dcd/dt=(cd/H)v where
CDi+1=CDi +h(CD*v/H) where initial condition Cd(0)=0.2(given)

Answer 7

wow..what class is this question for?..I"m taking an intermediate mechanics class and I have similar problems to that

Answer 8

numerical methods with matlab for civil engineers

Answer 9

cool