Question

in three different scenarios, a student of mass 3m pulls on a lightweight rope attached to a block of mass m. the student and the block are both initially at rest in each scenario. in scenario 1, the student stands on the shore of a frozen lake and the block is on the ice. after the student starts pulling on the rope in scenario 1, the block slides with negligible friction while the student remains in the same location on the shore. in scenario 2, both the student and the block are on the ice. after the student starts pulling on the rope in scenario 2, both the student and the block slide with negligible friction. in scenario 3, the block is at rest on the shore while the student is on the ice. after the student starts pulling on the rope in scenario 3, the student slides with negligible friction while the block remains in the same location on the shore. the final velocity of the block relative to the student is the same in each scenario. for which scenario, if any, is the final speed of the center of mass of the block - student system the greatest?
a scenario 1
b scenario 2
c scenario 3
d the speed of the center of mass is the same in all three scenarios.

Explanation:

Step1: Analyze the external forces

In each scenario, the only forces acting on the block - student system are internal forces (the tension in the rope). There are no external forces (since friction is negligible).

Step2: Apply the law of conservation of momentum

The law of conservation of momentum states that for a system with no external forces, the total initial momentum \(P_{i}\) equals the total final momentum \(P_{f}\). Initially, \(P_{i}=(3m + m)\times0 = 0\). So, \(P_{f}=(3m + m)v_{cm}=0\), where \(v_{cm}\) is the velocity of the center of mass.

Answer:

D. The speed of the center of mass is the same in all three scenarios.