Classical Mech - Newtons 2nd. Quad Air Resistance

[MPKU]

Homework Statement

A puck of mass m is kicked up an incline (angle θ) with initial speed v_o. Friction is not present, but air resistance has a magnitude of f(v) = cv². Solve Newtons second law for the pucks velocity as a function of t on the upward journey. How long does the journey last?

Homework Equations

The Attempt at a Solution

mr'' = -mgsinθ - f_quad

mv' = -mgsinθ - cv² v(hat)

dv/dt = -gsinθ -(c/m)v² v(hat)

dt = -dv/ (gsin θ -(c/m)v² v(hat) )


I'm not quite sure how to solve this; perhaps I could rewrite to get a known integral of 1/(1 +x^2) dx, but I don't see how.

 

[haruspex]

You can drop the v(hat), since you've reduced it to scalars, all motion being in the one dimension.
Can you do it from there?

 

[MPKU]

I don't think so. Should I just rewrite it as:

dt = -(gsin θ -(c/m)v^2)^-1 dv and integrate?

 

[haruspex]

I don't think so. Should I just rewrite it as:

dt = -(gsin θ -(c/m)v^2)^-1 dv and integrate?

Yes, except that you just made a sign error.