A textbook of mass m = 1.24 kg starts at rest on a frictionless inclined plane (

Question

A textbook of mass m = 1.24 kg starts at rest on a frictionless inclined plane (angle theta = 30o).  Although there is no friction, suppose there is a drag force (due to air resistance) acting on the book which is proportional to the speed squared and is described by the equation F=kmv^2, where k = 0.86 1/m.  How much time does it take for the textbook to slide a distance d = 1.65 m down the plane?    (Hint: This one is tricky, you will need to solve the integral by hand using a hyperbolic trig substitution.)

Explanation / Answer

Force acting on book = mg vertical downward

Component of mg along incline = mgSin30

Force balance:

mgSin30 - Fdrag = ma

=> mgSin30 - kmv^2 = ma

=> 5 - 0.86 v^2 = a

a = dv/dt

=> 5 - 0.86 v^2 = dv/dt

=> dv/(5-0.86v^2) = dt

Integrating on both sides:

(1/0.86)*ln[(5+0.86v)/(Sqrt(25 - (0.86v)^2)] = t

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