Two bodies of masses, `m_A` and `m_B`, are thrown up simultaneously. With reference to this situation, which of the given statement(s) about these bodies is/are correct?
I. Both bodies fall towards the earth with same acceleration.
Il. The body with heavier mass falls with more acceleration.
III. The acceleration of the bodies in this case is directly proportional to the mass.

Options

A. Only I
B. Only III
C. I and III
D. II and III

Answer

To solve the problem, we need to evaluate the correctness of the three statements regarding the motion of two bodies of different masses, \( m_A \) and \( m_B \), when they are thrown upwards.

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Bodies**:
- When the bodies are thrown upwards, they experience a gravitational force pulling them back towards the Earth. This force is given by Newton's law of gravitation, which states that the gravitational force \( F \) on a mass \( m \) due to the Earth (mass \( M \)) is:
\[ F = \frac{G \cdot M \cdot m}{r^2} \]
- Here, \( G \) is the universal gravitational constant, and \( r \) is the distance from the center of the Earth to the mass.

2. **Applying Newton's Second Law**:
- According to Newton's second law, the force is also equal to mass times acceleration:
\[ F = m \cdot a \]
- Setting these two expressions for force equal gives:
\[ m \cdot a = \frac{G \cdot M \cdot m}{r^2} \]
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ a = \frac{G \cdot M}{r^2} \]
- This shows that the acceleration \( a \) due to gravity is independent of the mass of the object.

3. **Evaluating Each Statement**:
- **Statement I**: "Both bodies fall towards the earth with the same acceleration."
- This statement is **correct** because we derived that the acceleration due to gravity is independent of the mass of the body.

- **Statement II**: "The body with heavier mass falls with more acceleration."
- This statement is **incorrect** because, as shown, the acceleration is the same for both masses regardless of their weight.

- **Statement III**: "The acceleration of the bodies in this case is directly proportional to the mass."
- This statement is **incorrect** because the acceleration is independent of the mass of the bodies.

4. **Final Conclusion**:
- The only correct statement is Statement I. Therefore, the answer to the question is that only Statement I is true.

### Summary of Correctness:
- **Statement I**: Correct
- **Statement II**: Incorrect
- **Statement III**: Incorrect

### Answer:
Only Statement I is correct.

---