# 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

Fd=CdApav2/2
Where
Fd is the drag force
Cd is the drag coefficient
A is the area exposed to the fall
pa the air density
v the immediate velocity

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

Fg=mg

## The Attempt at a Solution

ma=mg - mgpa/pc - CdApav2/2
Or
dv/dt = A - Bv2
Where
A=g( 1 - pa/pc)
B=CdApa/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