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Step-by-step Solution
Step 1: Identify the Key Concept
Because the man and the stone system is in a gravity-free space, there is no external force acting on the system. This implies the center of mass of the system remains at a fixed position throughout the motion (it does not move). We will use the principle of conservation of the center of mass.
Step 2: Define the Initial Conditions
• Mass of the man = $50\,\text{kg}$.
• Mass of the stone = $0.5\,\text{kg}$.
• The man (and stone, initially) is $10\,\text{m}$ above the floor.
Step 3: Determine the Displacements
When the stone reaches the floor (moves $10\,\text{m}$ downward), let the man’s final position be $10 + x$ meters above the floor, where $x$ is how much the man rises from his initial position.
Step 4: Apply the Center of Mass Conservation
The total mass times the initial center of mass height equals the total mass times the final center of mass height. But more simply, since the center of mass remains fixed and does not move, the displacement of the man will be such that the mass-weighted sum of displacements is zero. This can be written as:
$M_{\text{man}} \times x = M_{\text{stone}} \times 10$
Here, the stone’s displacement is $10\,\text{m}$ downward, so to keep the center of mass unchanged, the man must move upward sufficiently that the mass-displacement product is the same on both sides.
Step 5: Solve for x
$x = \frac{M_{\text{stone}}}{M_{\text{man}}} \times 10 = \frac{0.5}{50} \times 10 = 0.1\,\text{m}$
This means the man rises by $0.1\,\text{m}$.
Step 6: Find the Final Height of the Man
Initially, he was at $10\,\text{m}$. After rising by $0.1\,\text{m}$, the final height becomes:
$10 + 0.1 = 10.1\,\text{m}$
Step 7: Conclude the Answer
The distance of the man above the floor when the stone reaches the floor is $\boxed{10.1\,\text{m}}$.
