Vertical Motion with Quadratic Air Resistance

[Rumble]

Before I write anything, I want to apologize because I have no idea how to write equations on this website. This is my first post >.< Also, thank you for helping in advance!

Homework Statement

A baseball is thrown vertically up with speed v_o and is subject to a quadratic drag with magnitude f(v) = cv². Write down the equation of motion for the upward journey (measuring y vertically UP) and show that it can be rewritten as v(dot) = -g[1+(v/v_ter)²]. Use the "vdv/dx rule" to write v(dot) as vdv/dy and then solve the equation of motion by separating variables (put all terms involving v on one side and all terms involving y on the other). integrate both sides to give y in terms of v, and hence v as a function of y. Show that the baseball's maximum height is

y_max = [(v_ter)²/2g]*ln[ [ (v_ter)² + (v_o)² ] / [(v_ter)²] ]

whew. If v_o = 20m/s and the baseball has the parameters: mass m=.15kg and diameter D = 7cm, what is ymax? Compare with the value in a vacuum.

Homework Equations

Ok... Well first, in case you didn't get it, the vdv/dx rule is just that:

v(dot) = vdv/dx = (1/2)d(v²)/dx.

(only in this problem we just use y instead of x.)

Another formula that's important is the terminal velocity, which is
v_ter = sqrt(mg/c)

The Attempt at a Solution

Well, the first thing it asks is to write down the equation of motion. I'm a little unsure, but I think that it is :

m*v(dot) = -mg - cv²

which can be rearranged:
v(dot) = -g - cv²/m

and substituting c/m = g/(v_ter)² in...
v(dot) = -g (1 + (v/v_ter)²)

so then we use the vdv/dx rule...
vdv = -g*dy*(1 + (v/v_ter)²)

and separating variables like it said,
vdv/(1 + (v/v_ter)²) = -gdy

But now I'm not sure what I'm supposed to do. When it said to separate variables, it said that I should put the terms with a y on one side and the terms with a v on the other, but... are there any terms with a y? Other than the dy? I also have no idea how to integrate this equation... Can anybody help me figure out the next few steps? Thank you again.

PS: is there a way to actually have it write v(dot) normally - as in, with a dot above the v?

 

[Rumble]

Ok so I went ahead and tried to integrate vdv/(1 + (v/v_ter)²) = -gdy

On the left, I went from v_o to v and on the right i went from 0 to y. This gave me (and watch out i switch the left and right sides here):

-gy = sqrt[ (1/v_ter²)*v² + 1 ] / (1/v_ter²) from v_o to v.

Now, v_o = 0 at y_max, so you can plug those in, and you get

-gy_max = (v_ter)²*sqrt[ (1/v_ter²)*v² + 1 ] from v_o to 0

which simplifies to

-gy_max = v_ter² * (1 - sqrt[ (v_o/v_ter)² +1 ]

and I have no idea how to make that into the given equation (with ln and stuff) that is shown in my first post.

 

[haruspex]

vdv/(1 + (v/v_ter)²) = -gdy
-gy = sqrt[ (1/v_ter²)*v² + 1 ] / (1/v_ter²)

No, that integration step with v is wrong. Please write it out in more detail.