91Ó°ÊÓ

The concept of conservation of momentum is fundamental in physics, especially when analyzing systems where internal forces are at play. In an isolated system, like the man and the boat, the total momentum remains constant as long as no external forces are acting on it.
Imagine momentum as a token of motion that's distributed throughout the system. If the man moves one way, the boat must move in the opposite direction to balance things out. This ensures that the total momentum of the system, originally at rest, remains zero.
To find the displacement of the boat when the man walks across it, we use the conservation of momentum principle. The formula we use here is:
\[ \text{Displacement of boat} = \frac{m}{M+m} L \]
This formula arises from set conditions in the problem, ensuring that the motion caused by the man is counteracted precisely by the motion of the boat.

Relative motion refers to the movement of an object as observed from another moving object. In the context of our problem, it's about understanding how the man's motion across the boat is perceived.
The man walks the entire length of the boat, and we need to consider both his motion and the boat's motion. From a fixed point outside the system, both the man and the boat are moving because the boat drifts backward as the man walks forward.
However, the man's relative displacement on the boat—essentially how far he walks with respect to the moving boat—equals the original length of the boat, \(L\).
This means while the boat compensates by moving in the opposite direction, together they result precisely in the man covering the length \(L\), which reflects the understanding of relative motion in this scenario.

Equilibrium in physics often refers to a state where all forces and motions are balanced. In this exercise, equilibrium is maintained for the center of mass of the system.
As the man walks from one end of the boat to the other, the system internally adjusts to prevent the center of mass from shifting. Since no external forces act, the center stays unchanged, echoing a static balance.
The boat moves in the opposite direction as the man walks, ensuring the center remains stationary. This behavior epitomizes the idea of equilibrium:

• The center of mass remains constant.
• Both man and boat dynamically shift to maintain balance.

This teaches us an essential physics lesson, that even in motion, systems strive for equilibrium if unperturbed by outer influences.