Differential equation for grav, boyant and drag force

[Agustin]

Homework Statement

So there is a falling object, you have to take into account the boyant force, the pull of gravity and the drag force
A time dependent distance equation is what we're looking for

Homework Equations

F_d=C_dAp_av²/2
Where
F_d is the drag force
C_d is the drag coefficient
A is the area exposed to the fall
p_a the air density
v the immediate velocity

F_b=mgp_a/p_c
F_b is the boyant force
mg is the weight of the object
p_c is the object's density
Note ( this equation is found from the original equation F_b=Volume submerged x air density x gravity; where the submerged volume is m/p_c)

F_g=mg

The Attempt at a Solution

ma=mg - mgp_a/p_c - C_dAp_av²/2
Or
dv/dt = A - Bv²
Where
A=g( 1 - p_a/p_c)
B=C_dAp_a/2m

I solve for v

v (t) = (A/B)^1/2 (1 + C₁e^-2(BA)1/2t)/( 1 - C₁e^-2(BA)1/2t)

Where C₁ is some arbitrary constant

Integrating we get the distance formula:

X (t) = (A/B)^1/2t+(1/B)ln( 1 - C₁e^{-2 (BA)1/2t}) + C₂
I don't know wether it's correct or not. I've used techniques i found on the internet for the integration. http://www.freemathhelp.com/forum/threads/47073-integral-(1-(e-x-1))-dx-Using-Partial-Fractions

 

[Geofleur]

See if it gives the right answers in special cases. For example, what does your result say for the case of no buoyant force and no drag force? What happens as $t \rightarrow \infty$, and what would you expect physically in this case? There are lots of things like this to look into.

Oh, and can you check that the resulting equation is dimensionally correct?