Univ. 7.74 (7.62 in 10th edition). 2 kg pckg, 53.1 deg incline, coeff kin frict .20, 4 m from spring with const 120 N/m. Speed just before hitting spring? How far compressed? How close to init pos on rebound?
Your solution didn't take account of the gravitational PE change during the compression of the spring.
Another way to set this up:
If x is the total distance between release of the object and max compression of the spring, and theta the angle of inclination, then the spring is compressed a distance of (x - 4 m) from its equilibrium position and we have
`dPE_grav = -m g x sin(theta)
`dW_friction_ON = -m g x cos(theta) * mu (where mu is the coeff of friction)
`dPE_spring = .5 k (x - 4 m)^2.
At the beginning and the point of max compression the KE is zero.
Work-energy tells us that
`dW_noncons_ON = `dKE + `dPE so that
`dW_friction_ON = `dPE_spring + `dPE_grav
Substituting the above expressions we can solve for x in terms of known quantities.