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## Homework Statement

A baseball is thrown vertically up with speed v

_{o}and is subject to a quadratic drag with magnitude f(v) = cv

^{2}. 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(v

^{2})/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 - cv

^{2}

which can be rearranged:

v(dot) = -g - cv

^{2}/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?