The girl begins to walk along the plank a constant velocity of 1.815 m/s to the top right relative to the plank, originally at rest, is free to slide on a frozen lake, which is flat, frictionless surface. The 54.45-kg girl standing on a 181.5-kg plank.
a. What is her velocity relative to the surface of the ice?
b. What is the velocity of the plank relative to the surface of the ice?
To find the velocities, we need to apply the principle of conservation of linear momentum. The total momentum of the system before the girl begins to walk is zero because both the girl and the plank are at rest. After the girl starts walking, the total momentum of the system will remain zero because there are no external forces acting on the system.
a. To find the velocity of the girl relative to the surface of the ice, we can consider that the girl and the plank move in opposite directions due to the conservation of momentum. Let's assume that the girl moves to the right, and the plank moves to the left with a velocity v.
According to the conservation of momentum, the momentum of the girl and the plank after the girl starts walking should cancel each other out. Therefore,
(mass of the girl) × (velocity of the girl) + (mass of the plank) × (velocity of the plank) = 0
Substituting the given values:
(54.45 kg) × (1.815 m/s) + (181.5 kg) × (v) = 0
Now, solve the equation for v:
(54.45 kg) × (1.815 m/s) + (181.5 kg) × (v) = 0
987.96775 kg·m/s + 181.5 kg · v = 0
181.5 kg · v = -987.96775 kg·m/s
v = -987.96775 kg·m/s / 181.5 kg
v ≈ -5.43 m/s
The negative sign indicates that the velocity of the plank is in the opposite direction to the girl's velocity. Therefore, the girl's velocity relative to the surface of the ice is approximately 1.815 m/s to the right.
b. To find the velocity of the plank relative to the surface of the ice, we have already determined that the velocity of the plank is approximately -5.43 m/s.