Classical Mech - Newtons 2nd. Quad Air Resistance

AI Thread Summary
The discussion focuses on solving Newton's second law for a puck moving up an incline under the influence of air resistance, modeled as f(v) = cv². The equation of motion is derived as mr'' = -mgsinθ - fquad, leading to a differential equation for velocity as a function of time. Participants express uncertainty about integrating the equation, with suggestions to rewrite it for easier integration. A sign error is noted in one of the proposed solutions, emphasizing the importance of careful mathematical manipulation. The conversation highlights the complexities of applying Newton's second law in the presence of quadratic air resistance.
MPKU
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Homework Statement



A puck of mass m is kicked up an incline (angle θ) with initial speed vo. 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 - fquad

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

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

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