Work-Energy: Find the angle of the inclined plane

[epr2008]

Homework Statement

A block with mass 5 kg slides down and inclined plane with height 1.5 m and coefficient of kinetic friction .35. The final speed of the block is 4.2 m/s. Find the angle of the inclined plane.

Homework Equations

$${W^{{\rm{net}}}} = {W^{{\rm{grav}}}} - {W^{{\rm{fric}}}} = {K_2} - {K_1}$$

The Attempt at a Solution

This has taken me forever and i still don't think it's right but here it goes.

I have

$${W^{{\rm{net}}}} = mg(\Delta y) - {\mu _k}mgd\cos (\varphi ) = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$$

Then since initial velocity is 0

$${W^{{\rm{net}}}} = mg(\Delta y) - {\mu _k}mgd\cos (\varphi ) = \frac{1}{2}mv_2^2$$

I'm sure there is another way to find d but i couldn't think of any besides the law of sines.

$$\frac{d}{{\sin (90)}} = d = \frac{{\Delta y}}{{\sin (\varphi )}}$$

So now I have

$${W^{{\rm{net}}}} = mg(\Delta y) - {\mu _k}mg(\Delta y)\cot (\varphi ) = \frac{1}{2}mv_2^2$$

And solving

$$\cot (\varphi ) = \frac{{mg(\Delta y) - \frac{1}{2}mv_2^2}}{{{\mu _k}mg(\Delta y)}}$$

I am using the calculator on my phone so I don't know if it is right but

$$\varphi = 41.186^\circ$$

Would anyone care to help me out and tell me if I'm at least in the ballpark?

 

[Doc Al]

Looks good to me.