Free body diagram: F buoyancy = m * V *g, where V is the volume of the object an

Question

Free body diagram:

 

Fbuoyancy=m*V*g, where V is the volume of the object and equal to /6 *D

D is the diameter of the falling oject

Fgravitation =-*g*V

Fdrag=-3Dv, where v is the velocity of the particle: dy/dt.

 

Setup and Sovle the second order linear ODE with the following initial conditions:

Y(0)=y0 and V(o)=v0 

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I attempted to solve the following way:

Sum of forces =ma,  m*V*g-3DV-*g*V=ma  (1)

a=y'' and v=y'   (2)

replacing (2)->(1) we have : mVg-3Dy'-gV=my''

rearranging terms: my''+3Dy'=gV-mVg     (you can also conside V=m)

Solving for the homogeneous solution I found Yc=C1+C2e3Dt/V

and found C1=y0-V/3D and C2= v0(V/3D)

I know that these answers are incorrect because the dimensions are off.

Thank you for your help in advance(!!)

Free body diagram: Fbuoyancy=rhom*V*g, where V is the volume of the object and equal to pi/6 *D D is the diameter of the falling oject Fgravitation =-rho*g*V Fdrag=-3pimuDv, where v is the velocity of the particle: dy/dt. Setup and Sovle the second order linear ODE with the following initial conditions: Y(0)=y0 and V(o)=v0 --- I attempted to solve the following way: Sum of forces =ma, rhom*V*g-3pimuDV-rho*g*V=ma (1) a=y'' and v=y' (2) replacing (2)- > (1) we have : rhomVg-3pimuDy'-rhogV=my'' rearranging terms: my''+3pimuDy'=rhogV-rhomVg (you can also conside rhoV=m) Solving for the homogeneous solution I found Yc=C1+C2e3piDmut/Vrho and found C1=y0-Vrho/3pimuD and C2= v0(Vrho/3pimuD) I know that these answers are incorrect because the dimensions are off.

Explanation / Answer

m = V

mdv/dt = mgV - mg - 3Dv

dv/dt = mg/ - g - (3D/m)v

let c = mg/ - g = g(m/ - 1), b = 3D/m

dv/dt = c - bv, dv/(c - bv) = dt

integrate and use v(0) = v0

v = c/b - (c/b - v0)e-bt = dy/dt

integrate, and use y(0) = y0

y = y0 + ct/b + (c/b2 - v0/b)(1 - e-bt)

y = y0 + mgt(m/ - 1)/(3D) + [(m/ - 1)gm2/(3D)2 - mv0/(3D)](1 - e-3Dt/m)

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