(A) : A rocket works on the principle of conservation of linear momentum.
(r) : When there is no external force on a system of two bodies the rate of change in momenta of two bodies are equal and opposite

Options

A. Both (A) and (R) are true and (R) is the correct explanation of (A)
B. Both (A) and (R) are true and (R) is not the
C. (A) is true but ( R) is false
D. Both (A) and (R ) are false

Answer

To solve the question, we need to analyze the two statements provided:

**Statement A**: A rocket works on the principle of conservation of linear momentum.

**Statement R**: When there is no external force on a system of two bodies, the rate of change of momentum of the two bodies is equal and opposite.

### Step 1: Understanding Statement R
- According to Newton's second law, the net force acting on a system is equal to the rate of change of momentum. If there is no external force acting on the system, the net force is zero.
- Mathematically, this can be expressed as:
\[ F_{\text{net}} = 0 \implies \frac{dP_1}{dt} + \frac{dP_2}{dt} = 0 \]
- This implies that:
\[ \frac{dP_1}{dt} = -\frac{dP_2}{dt} \]
- This means that the rate of change of momentum of the first body is equal in magnitude but opposite in direction to that of the second body. Thus, Statement R is true.

### Step 2: Understanding Statement A
- A rocket operates based on the principle of conservation of linear momentum. When a rocket expels gas (fuel) downwards, it experiences an equal and opposite reaction that propels it upwards.
- Initially, the momentum of the system (rocket + fuel) is conserved. When the fuel is expelled, the momentum of the rocket increases in the upward direction while the momentum of the expelled fuel is in the downward direction.
- Mathematically, if the mass of the rocket is \(M_1\) and its velocity is \(V_1\), and the mass of the expelled fuel is \(M_2\) with velocity \(V_2\), the conservation of momentum can be expressed as:
\[ M_1 V_1 + M_2 (-V_2) = 0 \]
- This shows that the momentum before and after the fuel is expelled remains constant, confirming that Statement A is also true.

### Step 3: Relationship Between A and R
- Statement R provides a fundamental principle that explains why Statement A is true. The conservation of momentum in the rocket's operation is a direct application of the principle stated in R.
- Therefore, both statements are true, and R is the correct explanation of A.

### Conclusion
Both statements A and R are true, and R is the correct explanation of A.

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