The magnitude of gravitational potential energy of the moon earth system is U with zero potential energy at infinite separation. The kinetic energy of the moon with respect to the earth is K.

Answer

To solve the problem of finding the relationship between the gravitational potential energy (U) of the Moon-Earth system and the kinetic energy (K) of the Moon with respect to the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let \( M \) be the mass of the Moon.
- Let \( m \) be the mass of the Earth.
- Let \( A \) be the distance between the Moon and the Earth.
- Let \( V \) be the velocity of the Moon.

2. **Gravitational Potential Energy (U):**
- The gravitational potential energy (U) of the Moon-Earth system is given by the formula:
\[ U = -\frac{G M m}{A} \]
- Here, \( G \) is the gravitational constant.

3. **Kinetic Energy (K):**
- The kinetic energy (K) of the Moon is given by:
\[ K = \frac{1}{2} M V^2 \]

4. **Relate Velocity to Gravitational Force:**
- From Newton's law of gravitation, the gravitational force acting on the Moon is:
\[ F = \frac{G M m}{A^2} \]
- According to circular motion, this force provides the necessary centripetal force:
\[ F = \frac{M V^2}{A} \]
- Setting these two expressions for force equal gives:
\[ \frac{G M m}{A^2} = \frac{M V^2}{A} \]
- Simplifying this equation, we get:
\[ V^2 = \frac{G m}{A} \]

5. **Substitute Velocity in Kinetic Energy:**
- Substitute \( V^2 \) in the kinetic energy formula:
\[ K = \frac{1}{2} M \left(\frac{G m}{A}\right) \]
- This simplifies to:
\[ K = \frac{G M m}{2A} \]

6. **Relate U and K:**
- Now, we can relate the gravitational potential energy (U) to the kinetic energy (K):
\[ U = -\frac{G M m}{A} \]
- We see that:
\[ U = -2K \]

7. **Conclusion:**
- Since \( U \) is negative and \( K \) is positive, we can conclude that:
\[ |U| = 2K \quad \text{or} \quad U = -2K \]
- Therefore, the magnitude of gravitational potential energy \( |U| \) is greater than the kinetic energy \( K \).

### Final Relationship:
\[ |U| > K \]