Final answer:
To calculate the speed of each mass after one has fallen 1 meter, use the conservation of energy and Newton's laws to find the acceleration and final velocity, which ends up being 3.13 m/s for both masses.
Step-by-step explanation:
The problem of a 1.0 kg mass and a 2.5 kg mass on either side of a frictionless pulley is one that involves the principles of mechanics and conservation of energy. Since both masses are initially at rest and there's no friction involved with the pulley, we can assume that the only forces acting on the masses are due to gravity. The acceleration of the system and the final speed of the masses can be determined using Newton's second law and energy conservation respectively.
First, we calculate the acceleration of the system using Newton's second law (F=ma) where F is the net force and a is the acceleration. The net force is the difference in weights of the two masses (m2g - m1g). This gives us the equation a = (m2 - m1)g / (m1 + m2). Substituting the values, we get a = (2.5 kg - 1.0 kg) * 9.8 m/s² / (1.0 kg + 2.5 kg) = 4.9 m/s².
Since both masses start at rest, we use the equation for the final velocity V² = U² + 2as, where U is the initial velocity (0 m/s in this case), a is the acceleration we just calculated, and s is the distance (1.0 m). Plugging in the values gives V² = 0 + 2 * 4.9 m/s² * 1.0 m = 9.8 m²/s². Therefore, the final velocity V of each mass is √9.8 m²/s² = 3.13 m/s.