Final answer:
The change in the center-of-mass velocity of the system after a collision on a frictionless surface can be calculated using the conservation of momentum. In this case, a 1.0 kg mass moving at 8 m/s collides with a 3.0 kg mass at rest, and the change in velocity is found to be -6 m/s, indicating a reversal in direction.
Step-by-step explanation:
When considering a collision on a frictionless surface, the law of conservation of momentum is key to solving problems related to the center-of-mass velocity. In the scenario with masses A (1.0 kg) and B (3.0 kg), where Mass A is moving and Mass B is at rest, we can find the change in the center-of-mass velocity after they collide and stick together by applying this principle.
To find the initial momentum of the system, we consider only Mass A, as Mass B is initially at rest:
Initial momentum = (mass of A) × (velocity of A) = 1.0 kg × 8 m/s = 8 kg·m/s.
Since there are no external forces, the total momentum before and after the collision must be the same. Once the masses stick together, they will have a combined mass of 4.0 kg (1.0 kg + 3.0 kg), and they will move with a common velocity. Let's denote the final velocity as V.
Therefore, the final momentum of the system is the combined mass multiplied by the final velocity:
Final momentum = (combined mass) × (final velocity) = 4.0 kg × V kg·m/s.
By the conservation of momentum, the initial momentum equals the final momentum:
8 kg·m/s = 4.0 kg × V kg·m/s.
To find V, we divide both sides by 4.0 kg, which gives us:
V = 2 m/s.
Therefore, the change in the center-of-mass velocity as a result of the collision is the difference between the final and initial velocities of Mass A, which can be calculated as follows:
Change in center-of-mass velocity = Final velocity - Initial velocity of A = 2 m/s - 8 m/s = -6 m/s.
The negative sign indicates that the final center-of-mass velocity of the combined mass is in the opposite direction to Mass A's initial velocity.