Final answer:
The plank moves a distance of L/4 relative to the ground when the man walks from one end to the other end, using conservation of momentum on a frictionless surface. This is because the plank moves in the opposite direction at a speed three times faster than the speed of the man relative to the plank.
Step-by-step explanation:
The problem presented involves a man of mass m walking along a plank of length l on a frictionless surface. The plank has a mass of m/3. To solve this, we use the principle of conservation of momentum, which states that the total momentum of an isolated system remains constant if no external forces act upon it. Since there is no external horizontal force acting on the system (man plus plank), the center of mass of the system must remain stationary.
Let's define the direction the man walks as positive. Since the initial velocity of both man and plank is zero, the initial momentum of the system is zero. As the man walks with a velocity v relative to the plank, the plank moves in the opposite direction with a velocity V. Using conservation of momentum, for the system we have:
m * v = (m/3) * V
Solving for V gives us V = 3v. Thus, the plank moves three times faster than the man relative to the ground. If the man walks the entire length of the plank l, the plank will have moved a distance of l/4 relative to the ground, in the opposite direction, for the man to reach the end. The answer is (b) L/4.