Final answer:
To find the speed of the blocks after moving 2.0 m, the conservation of energy is applied, considering the initial potential energy lost by block 1 is converted into kinetic energy for both blocks. Since the system is frictionless, this calculation is straightforward, leaving us with a simple energy conservation equation to solve for speed.
Step-by-step explanation:
To find the speed of the blocks after they have each moved 2.0 m on a frictionless table and using a frictionless pulley, we can apply the principles of conservation of energy. Since the pulley has negligible mass and there is no friction, the only forces doing work are gravity on block 1 and the tension in the string, which does not do work as it pulls perpendicularly to the displacement.
Initially, the blocks are at rest, meaning their initial kinetic energy is zero. As block 1 falls a distance of 2.0 m, it loses potential energy, which is converted into kinetic energy for both blocks. The equation for gravitational potential energy is U = m₁gh, and the kinetic energy equation is KE = ½ mv².
Since U_initial = KE_final, using m₁ = 2.0 kg, m₂ = 4.0 kg, g = 9.8 m/s², and h = 2.0 m, we can set up the equation:
2.0 kg × 9.8 m/s² × 2.0 m = (½ × 2.0 kg + ½ × 4.0 kg) × v².
Simplifying and solving for v, we find the final speed of the blocks.
In this case, it's important to remember that both blocks move at the same speed due to the nature of the pulley system connecting them.