A body A of mass M while falling wertically downwards under gravity brakes into two parts, a body B of mass `(1)/(3)` M and a body C of mass `(2)/(3)` M. The center of mass of bodies B and C taken together shifts compared to that of body A towards

Options

A. does not shift
B. depends on height of breaking
C. body B
D. body C

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- We have a body A of mass M falling vertically under the influence of gravity. The center of mass of body A is at its geometric center.

2. **Division of Body A**:
- Body A breaks into two parts: body B with mass \( \frac{1}{3}M \) and body C with mass \( \frac{2}{3}M \).

3. **Finding the Center of Mass of the System**:
- The center of mass (CM) of a system of particles can be calculated using the formula:
\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
- In our case, we can consider the position of body A before it breaks apart as the origin (0,0) for simplicity.

4. **Position of Bodies After Division**:
- Let’s assume that at the moment of division, body B moves to position \( x_B \) and body C moves to position \( x_C \). Since we are only interested in the shift of the center of mass, we can analyze the positions relative to the center of mass of body A.

5. **Calculating the New Center of Mass**:
- The new center of mass of bodies B and C after the division can be calculated as:
\[ x_{CM} = \frac{\left(\frac{1}{3}M \cdot x_B\right) + \left(\frac{2}{3}M \cdot x_C\right)}{M} \]
- Since the center of mass of body A was at the origin, and the division occurs without any external force, the center of mass of the system of B and C will still be at the same point as body A.

6. **Conclusion**:
- Since the center of mass of bodies B and C coincides with the center of mass of body A before and after the division, we conclude that there is no shift in the center of mass of the combined system. Therefore, the center of mass of bodies B and C does not shift compared to that of body A.

### Final Answer:
The center of mass of bodies B and C taken together does not shift compared to that of body A.