The solid rubber balls A and B having masses 200 and 400 gm respectively are moving in opposite directions with velocity of A equal to 0.3 m / s . After collision the two balls come to rest, then the velocity of B is

Options

A. `0.15 ms^(-1)`
B. `-0.15 ms^(-1)`
C. `1.5 ms^(-1)`
D. `3.0 ms^(-1)`

Answer

To solve the problem, we will use the principle of conservation of momentum. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Mass of ball A, \( m_A = 200 \, \text{g} = 0.2 \, \text{kg} \)
- Mass of ball B, \( m_B = 400 \, \text{g} = 0.4 \, \text{kg} \)
- Velocity of ball A, \( v_A = 0.3 \, \text{m/s} \)
- Velocity of ball B, \( v_B = ? \) (to be determined)

2. **Understand the Direction of Motion:**
- Since the balls are moving in opposite directions, we can assign a negative sign to the velocity of ball B. Thus, we can denote \( v_B = -u \) where \( u \) is the magnitude of the velocity of ball B.

3. **Apply the Conservation of Momentum:**
- According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
- Before the collision:
\[ \text{Initial Momentum} = m_A \cdot v_A + m_B \cdot v_B \]
- After the collision, both balls come to rest, so:
\[ \text{Final Momentum} = 0 \]

4. **Set Up the Equation:**
\[ m_A \cdot v_A + m_B \cdot (-u) = 0 \]
Substituting the values:
\[ 0.2 \cdot 0.3 + 0.4 \cdot (-u) = 0 \]

5. **Solve for \( u \):**
\[ 0.06 - 0.4u = 0 \]
Rearranging gives:
\[ 0.4u = 0.06 \]
\[ u = \frac{0.06}{0.4} = 0.15 \, \text{m/s} \]

6. **Determine the Velocity of Ball B:**
Since we assigned a negative sign to \( v_B \):
\[ v_B = -u = -0.15 \, \text{m/s} \]

### Final Answer:
The velocity of ball B is \( -0.15 \, \text{m/s} \). The negative sign indicates that it is moving in the opposite direction to ball A.