A57 g ice cube can slide without friction up and down a 33 ∘ slope. the ice cube is pressed against a spring at the bottom of the slope, compressing the spring 12 cm . the spring constant is 20 n/m . when the ice cube is released, what distance will it travel up the slope before reversing direction?

The springs stored energy is transferred to the cube as kinetic energy and then by the slop the KE is converted to height energy. 

0.5 . k . x^2 = 0.5 . m . v^2 = m . g . ∆h 

0.5 . 50 . (0.1^2) = 0.05 . 9.8 . ∆h 

∆h = 0.51 m = 51 cm 

This is the height gained 
Distance along the slope = ∆h / sin 60 = 0.589 = 59 cm 

In the second case, the stored spring energy is converted into height energy AND frictional heat energy. 

The height energy is m . g . d sin 60 where d is the distance the cube moves along the slope. 

The Frictional energy converted is F . d 

F ( the frictional force ) = µ . N 

N ( the reaction to the component of the gravity force perpendicular to the surface of the slope ) = m . g . cos60 

Total energy converted 

0.5 . k . x^2 = (m . g . dsin60) + (µ . m . g . cos60 . d ) 

Solve for d 

d = 0.528 = 53 cm

26.4 is about the right time