Two skaters, one with mass 75 kg and the other with mass 40 kg, stand on an ice rink holding a pole of length 10 m and negligible mass. Starting from the ends of the pole. the skaters pull themselves along the pole until they meet. How far does the 40 kg skater move?

Answer

To solve the problem of how far the 40 kg skater moves when both skaters pull themselves along the pole until they meet, we can follow these steps:

### Step 1: Understand the System
We have two skaters with masses:
- Skater 1 (M1) = 75 kg
- Skater 2 (M2) = 40 kg

They are initially at the ends of a 10 m pole. We need to find out how far the 40 kg skater (M2) moves towards the 75 kg skater (M1).

### Step 2: Set Up the Coordinate System
Let’s set up a coordinate system along the pole:
- Position of M1 (75 kg) at x = 0 m (left end)
- Position of M2 (40 kg) at x = 10 m (right end)

### Step 3: Define the Displacements
Let:
- The distance moved by M2 (40 kg) towards M1 be x.
- The distance moved by M1 (75 kg) towards M2 will then be (10 - x) since the total length of the pole is 10 m.

### Step 4: Apply the Center of Mass Concept
Since there are no external forces acting on the system (the skaters are pulling themselves), the center of mass of the system will remain stationary. The formula for the center of mass (CM) of a two-particle system is given by:

\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

### Step 5: Calculate the Initial Position of the Center of Mass
Initially, the center of mass (CM) is calculated as follows:

\[ x_{CM} = \frac{(75 \, \text{kg} \cdot 0) + (40 \, \text{kg} \cdot 10 \, \text{m})}{75 \, \text{kg} + 40 \, \text{kg}} = \frac{0 + 400}{115} = \frac{400}{115} \approx 3.48 \, \text{m} \]

### Step 6: Set Up the Equation for Final Position of the Center of Mass
After they meet, the new positions will be:
- M1 will have moved (10 - x) m to the right.
- M2 will have moved x m to the left.

The new position of the center of mass will still be:

\[ x_{CM} = \frac{(75 \cdot (10 - x)) + (40 \cdot x)}{115} \]

### Step 7: Set the Two Center of Mass Equations Equal
Since the center of mass does not change position:

\[ \frac{(75 \cdot (10 - x)) + (40 \cdot x)}{115} = \frac{400}{115} \]

### Step 8: Solve for x
Multiply both sides by 115 to eliminate the denominator:

\[ 75(10 - x) + 40x = 400 \]

Expanding and simplifying:

\[ 750 - 75x + 40x = 400 \]
\[ 750 - 35x = 400 \]
\[ -35x = 400 - 750 \]
\[ -35x = -350 \]
\[ x = \frac{350}{35} = 10 \]

### Step 9: Calculate the Distance M2 Moves
Thus, the distance moved by the 40 kg skater (M2) is:

\[ x = 10 \, \text{m} \]

### Final Answer
The 40 kg skater moves 6.5 m towards the 75 kg skater.

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