To solve the problem step by step, we will use the principles of conservation of momentum and kinematics.
### Step 1: Identify the given data
- Mass of the man, \( m_1 = 50 \, \text{kg} \)
- Mass of the stone, \( m_2 = 0.5 \, \text{kg} \)
- Initial speed of the stone when thrown downwards, \( v_2 = 2 \, \text{m/s} \)
- Initial height of the man above the floor, \( h = 10 \, \text{m} \)
### Step 2: Apply the conservation of linear momentum
When the man throws the stone downwards, the momentum before and after throwing must be conserved.
The equation for conservation of momentum can be written as:
\[
m_1 \cdot v_1 + m_2 \cdot v_2 = 0
\]
Here, \( v_1 \) is the speed of the man moving upwards after throwing the stone.
Rearranging gives:
\[
m_1 \cdot v_1 = -m_2 \cdot v_2
\]
Substituting the values:
\[
50 \cdot v_1 = -0.5 \cdot 2
\]
\[
50 \cdot v_1 = -1
\]
\[
v_1 = \frac{-1}{50} = -0.02 \, \text{m/s}
\]
(The negative sign indicates that the man moves in the opposite direction to the stone.)
### Step 3: Calculate the time taken for the stone to reach the floor
The stone falls a distance of \( 10 \, \text{m} \) with an initial speed of \( 2 \, \text{m/s} \).
Using the formula:
\[
t = \frac{s}{v_2}
\]
Where \( s = 10 \, \text{m} \) and \( v_2 = 2 \, \text{m/s} \):
\[
t = \frac{10}{2} = 5 \, \text{s}
\]
### Step 4: Calculate the distance covered by the man in the opposite direction
The distance covered by the man while the stone is falling can be calculated using:
\[
\text{Distance} = v_1 \cdot t
\]
Substituting the values:
\[
\text{Distance} = -0.02 \cdot 5 = -0.1 \, \text{m}
\]
(The negative sign indicates that the man moves upwards.)
### Step 5: Calculate the final distance of the man above the floor
The final height of the man above the floor can be calculated as:
\[
\text{Final height} = \text{Initial height} + \text{Distance covered by the man}
\]
Substituting the values:
\[
\text{Final height} = 10 + 0.1 = 10.1 \, \text{m}
\]
### Conclusion
The distance of the man above the floor when the stone reaches the floor is \( 10.1 \, \text{m} \).
