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## 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

_{d}Ap

_{a}v

^{2}/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

_{d}Ap

_{a}v

^{2}/2

Or

dv/dt = A - Bv

^{2}

Where

A=g( 1 - p

_{a}/p

_{c})

B=C

_{d}Ap

_{a}/2m

I solve for v

v (t) = (A/B)

^{1/2}(1 + C

_{1}e

^{-2(BA)1/2t})/( 1 - C

_{1}e

^{-2(BA)1/2t})

Where C

_{1}is some arbitrary constant

Integrating we get the distance formula:

X (t) = (A/B)

^{1/2}t+(1/B)ln( 1 - C

_{1}e

^{-2 (BA)1/2t}) + C

_{2}

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