Projectile Motion with Quadratic Drag: Solving for Vertical Motion Equation

[gms7777]

Homework Statement

This is a small part of a problem I'm getting stuck on. I'm going to skip past the beginning steps to a part that I know is correct before I get stuck.
We have a baseball thrown up into the air, with quadratic drag. I'm attempting to solve for an equation of motion y(v).

Homework Equations

The Attempt at a Solution

We end up getting an equation

d/dt v = -g[1+(v/v_t)^2]

(I know that this is correct, the book gives it)

where v_t is the terminal velocity. We can then rewrite d/dt v = v dv/dy.

After separating variables, I get

v/[1+(v/v_t)^2] dv = -g dy

I integrate both sides and get

gy = 1/2 v_t^2 ln(v_t^2+v^2)+C

This is really the part where I'm confused. I thought that we can't take the logarithm of a non-unitless number and v_t^2+v^2 is definitely not unitless.

 

[gms7777]

Well, I solved my own problem. I redid the integral and ended up with

y = v_t^2/2g ln((v_t^2+v^2)/v_t^2) + c

Thanks anyway. :)