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## Homework Statement

A textbook of mass m = 1.24 kg starts at rest on a

frictionless inclined plane (angle = 30◦). 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 = kmv2, where k = 0.86

m−1. 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.)

## Homework Equations

Just F=m*a

drag force = k*m*v^2

## The Attempt at a Solution

So the mass keeps canceling on me whenever I run the problem, don't know if I actually need it for this one. I changed my coordinate system so positive x was parallel to the incline and positive y was perpendicular to it.

Forces are gravity broken in componets now, normal force only acting in positive y, canceled out by gravities y component. drag force in -x direction and +x component of gravity.

Vf= final velocity

y- direction F=ma

F= 0, its moving down the incline

x - direction F= ma

m*g*cos(theata)-k*m*v^2 = m dv/dt

solve it for time with given conditions and I got

T = (1/(Sqrt[g*sin(theata)*k])*ArcTanh[Sqrt[k/(g*sin(theata))]*Vf]

Switch the beginning equation to m*g*cos(theata)-k*m*v^2 = m*v* dv/dx and solve for the distance given the distance yields:

x = (1/2*k)*Ln((g*sin(theata)-k*Vf^2)/(g*sin(theata)))

Solve that equation for Vf yields:

Vf = Sqrt[(g*sin(theata)/k)*(1-e^(2*k*x))/k)

Sub into T equation and get:

T = (1/(Sqrt[g*sin(theata)*k])*ArcTanh(1-e^(2*k*x))

I keep getting very small times less than one and complex numbers which I feel are wrong, well its telling me they are wrong. I might have set it up wrong when defining my coordinate system or something. if someone can help me that would be awesome, even if its on a regular defined coordinate axes.