How Is Energy Conservation Used to Calculate Sliding Distance on an Incline?

[minimax]

Homework Statement

A block with a mass of 8.7kg slides down a hill with an angle of 27.4 degrees at an initial velocity of 1.66m/s. The friction uk between the block and hill is 0.62. What is the distance the block slides down the hill?

I solved the equation using components and kinetics, but I'm not quite sure how to solve it using energy conservation.

Homework Equations

For energy:
I know
initial energy=final energy
KE+PE= ? +ukNx ?

The Attempt at a Solution

Here is my work solving it with components
N(0,1)+mg(sin27.4, -cos27.4)+ukN (-1,0)=ma (1,0)
N=mgcos27.4

a=gsin27.4-ukmgcos27.4
=-.89m/s^2

Vf^2=Vi^2+2ad
d=-vo^2/2a
d=1.56m

hopefully doing it this way is also right. Thank you for any help!

 

[Shooting Star]

> a=gsin27.4-ukmgcos27.4

m shouldn't be there.

 

[minimax]

Oops, looks like i made a typo. but regardless of that the acceleration is still -0.89m/s^2

however, I'm still pondering on how to do this que with energy conservation.

KE-Wf=0?

 

[Shooting Star]

Initial KE + difference of PE_grav between initial and final positions = work done against friction = frictional force*distance.

 

[rootX]

KE_f - KE_i = Work by gravity + Work by friction
0 - KE_i = Work by gravity + Work by friction
- 0.5 m * (1.66)^2 = (m * 9.81 * sin 27.4 - m * 9.81 * 0.62 * cos 27.4)*ans
ans = 1.56 m

Sorry, I wouldn't be able to explain any concepts behind what I did: use geometry and FBDs.
But, this is the way you should be doing this; if you keep doing these questions
your way, you will be in a big mess one day (maybe on your exam day)