Two steel balls A and B of mass 10 kg and 10 g rolls towards each other with 5 m/s and 1 m/s respectively on a smooth floor. After collision, with what speed B moves (perfectly elastic collision)?

Options

A. `8 m/s`
B. `10 m/s`
C. `11 m/s`
D. Zero

Explanation

To solve the problem of finding the speed of ball B after a perfectly elastic collision with ball A, we will follow these steps:

### Step 1: Identify the masses and initial velocities
- Mass of ball A (m_A) = 10 kg
- Mass of ball B (m_B) = 10 g = 0.01 kg (convert grams to kilograms)
- Initial velocity of ball A (u_A) = 5 m/s (towards ball B)
- Initial velocity of ball B (u_B) = -1 m/s (towards ball A, hence negative)

### Step 2: Understand the concept of elastic collision
In a perfectly elastic collision, both momentum and kinetic energy are conserved. However, since ball A is significantly heavier than ball B, we can also use the property that the velocity of the heavier object (A) remains approximately unchanged after the collision.

### Step 3: Calculate the final velocity of ball A
Since ball A is much heavier than ball B, we can assume that its final velocity (v_A) remains approximately the same as its initial velocity:
- v_A ≈ u_A = 5 m/s

### Step 4: Use the conservation of momentum
The total momentum before the collision is equal to the total momentum after the collision:
\[ m_A \cdot u_A + m_B \cdot u_B = m_A \cdot v_A + m_B \cdot v_B \]
Substituting the known values:
\[ (10 \, \text{kg} \cdot 5 \, \text{m/s}) + (0.01 \, \text{kg} \cdot -1 \, \text{m/s}) = (10 \, \text{kg} \cdot 5 \, \text{m/s}) + (0.01 \, \text{kg} \cdot v_B) \]

### Step 5: Simplify the equation
Calculating the left side:
\[ 50 \, \text{kg m/s} - 0.01 \, \text{kg m/s} = 50 \, \text{kg m/s} + 0.01 \, \text{kg} \cdot v_B \]
This simplifies to:
\[ 49.99 \, \text{kg m/s} = 50 \, \text{kg m/s} + 0.01 \, \text{kg} \cdot v_B \]

### Step 6: Solve for v_B
Rearranging the equation to isolate \( v_B \):
\[ 49.99 - 50 = 0.01 \cdot v_B \]
\[ -0.01 = 0.01 \cdot v_B \]
Dividing both sides by 0.01:
\[ v_B = -1 \, \text{m/s} \]

### Step 7: Interpret the result
The negative sign indicates that ball B is moving in the opposite direction after the collision.

### Final Answer
The speed of ball B after the collision is 11 m/s in the direction opposite to its initial motion.

---