A 48.2 kg girl is standing on a 129 kg plank. The plank, originally at rest, is free to slide on a frozen lake, which is a flat, frictionless surface. The girl begins to walk along the plank at a constant velocity of 1.66 m/s relative to the plank. What is her velocity relative to the ice surface?
Hint: the center of mass of the plank will stay in the same place as the girl walks along it.
Compute the speed of the plank using this fact. Then do a vector addition of the plank velocity and the girl's velocity relative to the plank (which was given)
To determine the girl's velocity relative to the ice surface, we need to first analyze the motion of the plank and the girl separately.
Let's denote the girl's mass as mg (48.2 kg) and the mass of the plank as mp (129 kg). The girl's velocity relative to the plank is given as vgp (1.66 m/s).
Since the girl and the plank are on a frictionless surface, there are no external forces acting on them in the horizontal direction. According to Newton's third law, every action has an equal and opposite reaction. Therefore, the force exerted by the girl on the plank (Fgp) is equal in magnitude and opposite in direction to the force exerted by the plank on the girl.
Using Newton's second law, we can express these forces as:
Fgp = mg * ag
Fpg = mp * ap
where ag is the acceleration of the girl and ap is the acceleration of the plank.
Since the plank is at rest, its acceleration is zero: ap = 0.
The net horizontal force acting on the plank is zero because there are no external forces. Therefore, the force exerted by the girl on the plank must also be zero: Fgp = 0.
From the first equation, we can deduce that:
0 = mg * ag
Since the mass of the girl is nonzero, the acceleration of the girl is zero: ag = 0.
This means the girl is walking at a constant velocity relative to the plank (1.66 m/s). Since the plank is not moving (ap = 0), the girl's velocity relative to the ice surface (vgi) is the same as her velocity relative to the plank (vgp).
Therefore, her velocity relative to the ice surface is also 1.66 m/s.