How Does Air Resistance Affect a Bullet's Terminal Velocity?

[oswaler]

[SOLVED] Air resistance

Homework Statement

A bullet of mass m, diameter d, and muzzle velocity v0 is fired vertically upward from the ground. Assume that the drag can be expressed with Prandtl's air resistance forumla and that the drag coefficient, c, and air density, p, are constant. Find the expression for the terminal velocity.

Homework Equations

Prandtl's equation for drag: W=(1/2)cpAv^2

c is drag cofficient
p is air density
A is cross-section area of the bullet
v is velocity

The Attempt at a Solution

So it seems there are 2 forces acting on the bullet, gravity and drag. I tried to set up the equation of motion as
F=-mg-W
so F=-mg-(1/2)cpAv^2
so m(dv/dt)=-mg-(1/2)cpAv^2

after separation I have
dt=mdv/(-mg-(1/2)cpAv^2)

when I integrate the right side I get an imaginary answer, so something's wrong here.

Or, I could take m(dv/dt)=-mg-(1/2)cpAv^2 and set -mg-(1/2)cpAv^2=0 which would mean dv/dt=0 which is where the terminal velocity is hit, but then when I solve for v, again there's a negative in the square root so again I have an imaginary velocity.

I think I must be setting up the original equation wrong but I'm not seeing where the problem is. Any help would be appreciated.

 

[natugnaro]

I think that dv/dt is never zero while the bullet is moving up.

 

[oswaler]

dv/dt=0 when the bullet is at the apex.