A 64-kg woman stands on friction less level ice with a 0.9-kg stone at her feet. She kicks the stone with her foot so that she acquires a velocity of 9.2 cm/s in the forward direction. The speed acquired by the stone is:
To find the speed acquired by the stone, we can use the principle of conservation of momentum. The total momentum before the kick should be equal to the total momentum after the kick.
The momentum of an object is given by the product of its mass and velocity. We can calculate the total momentum before the kick:
Total Initial Momentum = (mass of woman * velocity of woman) + (mass of stone * velocity of stone)
Given:
mass of woman = 64 kg
velocity of woman = 0 m/s (initially at rest)
mass of stone = 0.9 kg
velocity of stone = 0 m/s (initially at rest)
Total Initial Momentum = (64 kg * 0 m/s) + (0.9 kg * 0 m/s) = 0 kg.m/s
The total momentum after the kick should also be equal to 0 kg.m/s because the woman is on frictionless ice, meaning there are no external forces acting on the system.
Total Final Momentum = (mass of woman * final velocity of woman) + (mass of stone * final velocity of stone)
We need to find the final velocity of the stone. Since the woman is standing on frictionless ice, her momentum does not change. Therefore, the final velocity of the woman will still be 0 m/s.
Final velocity of woman = 0 m/s
Using the principle of conservation of momentum:
Total Initial Momentum = Total Final Momentum
0 kg.m/s = (64 kg * 0 m/s) + (0.9 kg * final velocity of stone)
Simplifying the equation:
0 = 0 + 0.9 kg * final velocity of stone
Therefore, the final velocity of the stone, which represents its speed, will be 0 m/s.
To find the speed acquired by the stone, we can use the principle of conservation of momentum.
The momentum before the kick is equal to the momentum after the kick.
The momentum of the woman before the kick is given by:
Momentum_woman = (Mass_woman) * (Velocity_woman)
The momentum of the stone before the kick is given by:
Momentum_stone = (Mass_stone) * (Velocity_stone)
Since the woman is initially at rest, her momentum before the kick is zero:
Momentum_woman = 0
Applying conservation of momentum, we have:
Momentum_woman + Momentum_stone = Momentum_woman_after + Momentum_stone_after
0 + (Mass_stone) * (Velocity_stone) = (Mass_woman) * (Velocity_woman_after) + (Mass_stone) * (Velocity_stone_after)
Since the woman is much heavier than the stone, we can assume that her velocity after the kick is negligible.
So, we have:
(Mass_stone) * (Velocity_stone) = (Mass_woman) * (0) + (Mass_stone) * (Velocity_stone_after)
Simplifying the equation, we get:
(Velocity_stone_after) = (Mass_stone) * (Velocity_stone) / (Mass_stone)
(Velocity_stone_after) = (Velocity_stone)
Therefore, the speed acquired by the stone after the kick is 9.2 cm/s.