Assertion: Two spherical bodies of mass ratio `1:2` travel towards each other (starting from rest) under the action of their mutual gravitational attraction. Then, the ratio of their kinetic energies at any instant is `2:1`
Reason: At any instant their momenta are same.

Options

A. If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
B. If both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.
C. If Assertion is true but Reason is false.
D. If Assertion is false but Reason is true.

Explanation

To solve the problem, we need to analyze the assertion and reason provided regarding the two spherical bodies with a mass ratio of 1:2 that are moving towards each other under their mutual gravitational attraction.

### Step-by-Step Solution:

1. **Understanding the Mass Ratio**:
- Let the masses of the two spherical bodies be \( m_1 \) and \( m_2 \) such that \( m_1:m_2 = 1:2 \).
- We can express this as \( m_1 = m \) and \( m_2 = 2m \).

2. **Initial Conditions**:
- Both bodies start from rest, meaning their initial velocities are zero.

3. **Gravitational Attraction**:
- As the bodies move towards each other, they experience mutual gravitational attraction, which causes them to accelerate. The gravitational force \( F \) acting on each body can be expressed as:
\[ F = \frac{G m_1 m_2}{r^2} \]
- However, for our analysis, we will focus on the kinetic energy rather than calculating the force directly.

4. **Momentum Conservation**:
- According to the law of conservation of momentum, the total momentum before they start moving is zero. As they move towards each other, their momenta must still sum to zero.
- Let \( v_1 \) and \( v_2 \) be the velocities of \( m_1 \) and \( m_2 \) respectively at any instant.
- Therefore, we have:
\[ m_1 v_1 + m_2 v_2 = 0 \implies m v_1 + 2m v_2 = 0 \implies v_1 = -2v_2 \]

5. **Kinetic Energy Calculation**:
- The kinetic energy \( KE \) of each body can be expressed as:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m v_1^2 \]
\[ KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} (2m) v_2^2 = m v_2^2 \]
- Substituting \( v_1 = -2v_2 \) into the kinetic energy of the first body:
\[ KE_1 = \frac{1}{2} m (-2v_2)^2 = \frac{1}{2} m (4v_2^2) = 2m v_2^2 \]

6. **Finding the Ratio of Kinetic Energies**:
- Now we can find the ratio of the kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{2m v_2^2}{m v_2^2} = \frac{2}{1} \]
- Thus, the ratio of their kinetic energies is \( 2:1 \).

7. **Conclusion**:
- The assertion that the ratio of their kinetic energies at any instant is \( 2:1 \) is correct.
- The reason that at any instant their momenta are the same is also correct, as shown by the conservation of momentum.

### Final Answer:
Both the assertion and reason are correct, and the reason is a correct explanation of the assertion.

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