# Block on Plane with Friction in strange coordinate system

## Homework Statement

A block of mass m slides down a plane inclined at an angle theta with initial velocity v down the slope and with friction coefficient mu. Find T, the time in which the block comes to a rest due to friction. Use coordinates with y vertical and x horizontal.

F=ma
f=(mu)n

## The Attempt at a Solution

$$\vec{v} = v \cos\theta \hat{x} - v \sin\theta \hat{y}$$

$$\\vec{\\F}_{w} = - m g \hat{y} \\$$

$$\vec{f} = - \mu mg \cos^{2}\theta \hat{x} + \mu m g \cos(\theta)\sin(\theta)\hat{y } \\$$

$$\ddot{x} = - \mu g \cos^{2}(\theta) \\$$

$$\dot{x} = - \mu g \cos^{2}(\theta) t + v\cos(\theta) \\$$

$$\ddot{y} = \mu g \cos(\theta)\sin(\theta) - g \\$$

$$\dot{y} = ( \mu g \cos(\theta)\sin(\theta) - g ) t - v \sin\theta \\$$

Now I should be able to set either y dot or x dot equal to zero and get T, right?
The answer I'm getting doesn't agree with the answer I got with the x axis down the plane and the y axis perpendicular to the plane. The answer I get in this way is

$$T = \frac {v_{0}}{g (\mu \cos \theta - \sin \theta)}$$

I hope this works; I've never used latex before.

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