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 1: Understand the System
We have two balls of mass \( m \):
- Ball 1 is thrown upward.
- Ball 2 is thrown downward.

Both balls are subject to the acceleration due to gravity, \( g \).

### Step 2: Define the Accelerations
- The acceleration of Ball 1 (upward) is \( -g \) (since upward is considered negative).
- The acceleration of Ball 2 (downward) is \( +g \).

### Step 3: Calculate the Acceleration of the Center of Mass
The formula for the acceleration of the center of mass \( a_{cm} \) for two masses \( m_1 \) and \( m_2 \) with accelerations \( a_1 \) and \( a_2 \) is given by:

\[ a_{cm} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2} \]

In our case:
- \( m_1 = m \) (mass of Ball 1)
- \( m_2 = m \) (mass of Ball 2)
- \( a_1 = -g \) (acceleration of Ball 1)
- \( a_2 = g \) (acceleration of Ball 2)

Substituting these values into the formula gives:

\[ a_{cm} = \frac{m(-g) + m(g)}{m + m} \]

### Step 4: Simplify the Expression
Now, simplify the equation:

\[ a_{cm} = \frac{-mg + mg}{2m} \]

This simplifies to:

\[ a_{cm} = \frac{0}{2m} = 0 \]

### Step 5: Conclusion
The acceleration of the center of mass of the system is \( 0 \). This means that the center of mass does not accelerate; it remains stationary in the absence of external forces.

### Final Answer
The acceleration of the center of mass is \( 0 \).

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