Final answer:
The block covers a distance of approximately 12.76 meters before coming to rest, calculated by equating the work done by friction to the initial kinetic energy of the block.
Step-by-step explanation:
To find the distance covered by the block until it comes to rest, we can use the work-energy principle. The work done by friction (negative work since it is opposite to the direction of motion) will equal the block's initial kinetic energy because it comes to rest (final kinetic energy is zero).
The work done by friction is given by:
Work = Friction Force × Distance = (μ_k × Normal Force) × Distance
The normal force for horizontal motion equals the weight of the block, which is:
Normal Force = mass × g
The initial kinetic energy of the block is:
Kinetic Energy = (1/2) × mass × (initial velocity)^2
Equating the work done by friction to the kinetic energy and solving for distance, we get:
Distance = (1/2) × mass × (initial velocity)^2 / (μ_k × mass × g)
Since the mass cancels out, our equation simplifies to:
Distance = (1/2) × (initial velocity)^2 / (μ_k × g)
Using the given values, initial velocity = 5 m/s, μ_k = 0.1, and g = 9.8 m/s^2, we calculate:
Distance = (1/2) × (5 m/s)^2 / (0.1 × 9.8 m/s^2) = 12.7551 meters
The block covers a distance of approximately 12.76 meters before coming to rest.