Two balls are thrown simultaneously in air. The acceleration of the centre of mass of the two balls while in air

Options

A. depends on the direction of the motion of the balls
B. depends on the masses of the two balls
C. depends on the speeds of the two balls
D. is equla to g

Answer

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

### Step 1: Understand the Concept of Center of Mass
The center of mass (COM) of a system of particles is the point where the total mass of the system can be considered to be concentrated. The acceleration of the center of mass can be determined by the accelerations of the individual masses in the system.

### Step 2: Identify the Forces Acting on the Balls
When the two balls are thrown into the air, the only force acting on them (after they are thrown) is the force of gravity. This force causes both balls to accelerate downwards at the acceleration due to gravity, denoted as \( g \).

### Step 3: Write the Formula for Acceleration of Center of Mass
The acceleration of the center of mass \( a_{cm} \) for a system of two particles can be expressed as:
\[ a_{cm} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2} \]
where \( m_1 \) and \( m_2 \) are the masses of the two balls, and \( a_1 \) and \( a_2 \) are their respective accelerations.

### Step 4: Substitute the Values
Since both balls are in free fall, their accelerations are equal to \( g \):
\[ a_1 = g \quad \text{and} \quad a_2 = g \]
Substituting these values into the formula gives:
\[ a_{cm} = \frac{m_1 g + m_2 g}{m_1 + m_2} \]

### Step 5: Factor Out the Common Term
We can factor out \( g \) from the numerator:
\[ a_{cm} = \frac{g(m_1 + m_2)}{m_1 + m_2} \]

### Step 6: Simplify the Expression
The \( m_1 + m_2 \) terms in the numerator and denominator cancel out:
\[ a_{cm} = g \]

### Conclusion
Thus, the acceleration of the center of mass of the two balls while in the air is equal to \( g \), which is the acceleration due to gravity.

### Final Answer
The acceleration of the center of mass of the two balls while in air is \( g \).

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