Two skaters A & B of mass M and 2M are standing together on a frictionless ice surface. They push each other apart. The skater B moves away from A with a speed of 2 m/s relative to ice. The separation between the two skaters after 5 seconds will be

Options

A. 30 m
B. 32 m
C. 40 m
D. 28 m

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Situation
We have two skaters, A and B, with masses M and 2M respectively, standing together on a frictionless ice surface. They push each other apart.

### Step 2: Identify the Given Information
- Mass of skater A = M
- Mass of skater B = 2M
- Speed of skater B (relative to ice) = 2 m/s
- Time after which we need to find the separation = 5 seconds

### Step 3: Apply Conservation of Momentum
Since there are no external forces acting on the system, the total momentum before and after the skaters push off each other must be conserved.

- Initial momentum (before they push off) = 0 (both are at rest)
- Final momentum = momentum of A + momentum of B

Let the velocity of skater A after they push off be \( V_A \) and the velocity of skater B be \( V_B = 2 \, \text{m/s} \).

Using conservation of momentum:
\[ 0 = M \cdot V_A + 2M \cdot V_B \]

### Step 4: Substitute Known Values
Substituting \( V_B = 2 \, \text{m/s} \):
\[ 0 = M \cdot V_A + 2M \cdot 2 \]
\[ 0 = M \cdot V_A + 4M \]

### Step 5: Solve for \( V_A \)
Rearranging the equation:
\[ M \cdot V_A = -4M \]
Dividing both sides by M (assuming M ≠ 0):
\[ V_A = -4 \, \text{m/s} \]

### Step 6: Determine the Relative Velocity
The relative velocity between skater A and skater B is:
\[ V_{\text{relative}} = V_A - V_B = -4 - 2 = -6 \, \text{m/s} \]
The negative sign indicates that they are moving apart.

### Step 7: Calculate the Separation After 5 Seconds
The distance (separation) between the two skaters after 5 seconds can be calculated using:
\[ \text{Distance} = \text{Relative Velocity} \times \text{Time} \]
\[ \text{Distance} = 6 \, \text{m/s} \times 5 \, \text{s} = 30 \, \text{m} \]

### Final Answer
The separation between the two skaters after 5 seconds will be **30 meters**.

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