Answer:
speed before reaching rough surface = 9.18 m/s
speed before hitting spring = 8.70 m/s
spring compression = 82 cm
number of complete trip = 9
Explanation:
Lets say
Position 1: On top of hill
Position 2: down the hill
Position 3: after the rough surface
Position 4: after hitting the spring
We'll strictly use conservation of energy for this equation
Potential energy on top of energy is full converted into kinetic energy down the hill (since surface is frictionless)
Hence, PEg1 = KE2
mgh = (1/2)mv2^2
(4.5)(9.8)(4.3) = (1/2)(4.5)v2^2
189.63 = (1/2)(4.5)v2^2
v2^2 = 2(9.8)(4.3) = 84.28
v2 = sqrt(84.28) = 9.18 m/s
After down the hill, it passes a rough surface. So some of the energy is loss due to friction forces
Friction force, Ff = u (coeff of kinetic friction ) x N (normal force)
Normal force, N = weight of box = mg = 4.5 x 9.8
Ff = 0.22 x 4.5 x 9.8
Work done / Energy loss = Wf = Ff x d (distance)
Wf = 0.22 x 4.5 x 9.8 x 2 = 19.404
Energy after passing the rough surface is totally kinetic energy
KE3 = KE2 - Wf = 189.63 - 19.404 = 170.226
speed after rough surface,
(1/2)mv3^2 = 170.226
v3 = sqrt((2 x 170.226)/4.5) = 8.70 m/s
After hitting the spring, all the kinetic energy is converted into potential energy of spring
170.226 = (1/2)kx^2
x^2 = 2 x 170.226 / 510 {note that constant of spring, k = 510}
x^2 = 0.668
x = sqrt(0.668) = 0.82m (82 cm)
To calculate complete trip before the box coming to rest, note that the only place where it loss energy is at the rough surface.
Energy before the first time pass rough surface = 189.63
Energy loss each time passing rough surface = 19.404
189.63 / 19.404 = 9.773 (9 complete with balance of 0.773)
That mean, the box will pass the rough surface 9 complete trip before coming to rest
