A ball of mass m is thrown upward and another ball of same mass is thrown downward so as to move freely gravity. The acceleration of centre of mass is

Options

A. g
B. `(g)/(2)`
C. 2g
D. Zero

Answer

To find the acceleration of the center of mass of the two balls, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the masses and their accelerations**:
- Let the mass of each ball be \( m \).
- The first ball is thrown upward, and its acceleration due to gravity is \( -g \) (downward).
- The second ball is thrown downward, and its acceleration is also \( -g \) (downward).

2. **Write down the formula for the acceleration of the center of mass**:
- The acceleration of the center of mass \( a_{cm} \) for a system of particles is given by:
\[ a_{cm} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2} \]
- Here, \( m_1 = m \), \( a_1 = -g \) (for the ball thrown upward), \( m_2 = m \), and \( a_2 = -g \) (for the ball thrown downward).

3. **Substitute the values into the formula**:
- Substitute \( m_1 \), \( a_1 \), \( m_2 \), and \( a_2 \) into the formula:
\[ a_{cm} = \frac{m(-g) + m(-g)}{m + m} \]

4. **Simplify the equation**:
- Combine the terms in the numerator:
\[ a_{cm} = \frac{-mg - mg}{2m} = \frac{-2mg}{2m} \]
- This simplifies to:
\[ a_{cm} = -g \]

5. **Interpret the result**:
- The negative sign indicates that the acceleration of the center of mass is directed downward, which means the acceleration of the center of mass is \( g \) downward.

### Final Answer:
The acceleration of the center of mass is \( g \) downward.

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