Final answer:
When the ice cube is released, it will travel up the slope until it reaches the maximum compression of the spring. The total distance traveled by the ice cube before reversing direction is approximately 0.4 meters. The plastic cube will travel a total distance of 0.2 meters up the slope.
Step-by-step explanation:
When the ice cube is released, it will travel up the slope until it reaches its maximum compression of the spring. The spring force acting on the ice cube can be calculated using Hooke's law: F = kx, where F is the force, k is the spring constant, and x is the compression distance. In this case, the force is equal to the weight of the ice cube: F = mg. Thus, we have the equation: mg = kx.
The distance traveled up the slope can be calculated using the work-energy theorem: W = ΔPE = Fd, where W is the work done, ΔPE is the change in potential energy, F is the force, and d is the distance. In this case, the work done is equal to the change in potential energy of the ice cube: mgd = 0.5kx². Solving for d, we get: d = (0.5k/m) * x².
Substituting the given values, we have: d = (0.5 * 28 N/m / 0.065 kg) * (0.1 m)² = 0.4 m.
Therefore, the ice cube will travel a total distance of 0.4 meters up the slope before reversing direction.
To find the distance traveled by the plastic cube, we need to consider the kinetic friction acting on it. The force of kinetic friction can be calculated using the equation: f_k = μ_k * N, where f_k is the force of kinetic friction, μ_k is the coefficient of kinetic friction, and N is the normal force. In this case, the normal force is equal to the weight of the plastic cube: N = mg.
The work done against friction can be calculated using the equation: W = f_k * d, where W is the work done, f_k is the force of kinetic friction, and d is the distance. Solving for d, we get: d = W / f_k.
Substituting the given values, we have: d = (W / (μ_k * mg)). Since the work done against friction is equal to the change in kinetic energy, we have: W = ΔKE = 0.5mv². Rearranging for v² and substituting the given values, we get: v² = (2W / m). Therefore, the distance traveled by the plastic cube can be calculated as: d = (2W / (μ_k * mg)).
Since the force required to compress the spring is the weight of the ice cube, we can use the equation for the work done against friction to find the work done against friction for the plastic cube: W = f_k * d = μ_k * N * d = μ_k * mg * d. Substituting this into the equation for d, we get: d = (2(μ_k * mg * d) / (μ_k * mg)) = 2d.
Therefore, the plastic cube will travel a total distance of 2 times the distance compressed by the spring, or 2 * 0.1 m = 0.2 m.