ODE water drop modeling questionI have been working on a ODE homework which involves modeling the velocity of a drop of water falling from the sky. The ODE that models its velocity is given by: m v ′ = k v 2 − m g , k = 1 2 C d ρ a , C d : friction, ρ a : air density, A: transversal section of the water drop.I have had to find the theoretical velocity limit of the water drop as a function of A by solving directly the ODE and compare these results with Euler and Runge-Kutta IV methods on MATLAB. I have done all of that.The last question of my homework is an open question: It asks to modify the ODE presented by imagining now that the water drop losses a fraction of mass (by evaporation) while it is falling. I will have to apply Euler and Runge-Kutta IV on this new ODE. So I am looking for suggestions to improve the equation. Mass m now is going to be a function of time, it has to decrease. I have been thinking to assume that the water drops are spheres and by using d e n s i t y ∗ v o l u m e = m a s s

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ODE water drop modeling questionI have been working on a ODE homework which involves modeling the velocity of a drop of water falling from the sky. The ODE that models its velocity is given by:  m      v    ′    =  k      v    2    −  m  g  ,    k  =      1    2        C          d            ρ          a        ,      C          d      : friction,       ρ          a      : air density, A: transversal section of the water drop.I have had to find the theoretical velocity limit of the water drop as a function of A by solving directly the ODE and compare these results with Euler and Runge-Kutta IV methods on MATLAB. I have done all of that.The last question of my homework is an open question: It asks to modify the ODE presented by imagining now that the water drop losses a fraction of mass (by evaporation) while it is falling. I will have to apply Euler and Runge-Kutta IV on this new ODE. So I am looking for suggestions to improve the equation. Mass m now is going to be a function of time, it has to decrease. I have been thinking to assume that the water drops are spheres and by using   d  e  n  s  i  t  y  ∗  v  o  l  u  m  e  =  m  a  s  s

RamPatWeese2w · Accepted Answer

Step 1I think you should just be able to write that the mass m is a function of time and perhaps assume a linear one, namely  m  (  t  )  =      m    0    −  s  t  ,where       m    0   is an initial mass and s is the rate of loss (positive) in units of             m      a      s      s              t      i      m      e      .Step 2Then your equation is no longer separable (but who cares if you&#039;re going to solve in numerically) and would look like      v    ′    =      k          m      (      t      )            v          2        −  g  ,or      v    ′    =      k                  m        0            −      s      t            v          2        −  g  ,

I have been working on a ODE homework which involves modeling the velocity of a drop of water falling from the sky. The ODE that models its velocity is given by: mv'=kv^2-mg, k=1/2 C_d rho_a

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