A Body moving radially away from a planet of mas M, when at distance r from planet, explodes in such a way that two of its many fragments move in mutually prependicular circular orbits around the planet what will be
(i). Then velocity in circular orbits?
(ii). Maximum distance between the two fragments before collision and
(iii). Magnitude of their relative velocity just before they collide?

Explanation

To solve the problem step by step, we will address each part of the question sequentially.

### Given:
- A body is moving radially away from a planet of mass \( M \) at a distance \( r \) from the planet.
- The body explodes, and two fragments move in mutually perpendicular circular orbits around the planet.

### (i) Velocity in Circular Orbits

1. **Understanding Circular Motion**: The velocity \( v \) of an object in circular motion around a planet can be derived from the gravitational force acting as the centripetal force.

2. **Using the Formula**: The gravitational force provides the necessary centripetal force for circular motion:
\[ F = \frac{GMm}{r^2} = \frac{mv^2}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the fragment, and \( r \) is the radius of the circular orbit.

3. **Cancelling \( m \)**: Since \( m \) appears on both sides, we can cancel it:
\[ \frac{GM}{r^2} = \frac{v^2}{r} \]

4. **Rearranging the Equation**: Rearranging gives:
\[ v^2 = \frac{GM}{r} \]

5. **Taking the Square Root**: Therefore, the velocity \( v \) in circular orbit is:
\[ v = \sqrt{\frac{GM}{r}} \]

### (ii) Maximum Distance Between the Two Fragments Before Collision

1. **Understanding the Geometry**: The two fragments are moving in mutually perpendicular circular orbits. The maximum distance between them occurs when they are at the farthest points in their respective orbits.

2. **Calculating Maximum Separation**: If the radius of each orbit is \( r \), the maximum distance \( d \) between the two fragments is given by:
\[ d = r + r = 2r \]
However, since they are in perpendicular orbits, we can also use the Pythagorean theorem:
\[ d = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} \]

### (iii) Magnitude of Their Relative Velocity Just Before They Collide

1. **Understanding Relative Velocity**: The relative velocity of two objects moving in perpendicular directions can be calculated using the Pythagorean theorem.

2. **Calculating Individual Velocities**: The velocities of the two fragments in their respective orbits are both \( v = \sqrt{\frac{GM}{r}} \).

3. **Using the Formula for Relative Velocity**: The magnitude of the relative velocity \( V_{relative} \) is given by:
\[ V_{relative} = \sqrt{V_1^2 + V_2^2} \]
where \( V_1 \) and \( V_2 \) are the velocities of the two fragments.

4. **Substituting the Values**: Since both velocities are equal:
\[ V_{relative} = \sqrt{v^2 + v^2} = \sqrt{2v^2} = v\sqrt{2} \]

5. **Final Expression**: Substituting \( v = \sqrt{\frac{GM}{r}} \):
\[ V_{relative} = \sqrt{2} \cdot \sqrt{\frac{GM}{r}} = \sqrt{\frac{2GM}{r}} \]

### Summary of the Results:
- (i) Velocity in circular orbits: \( v = \sqrt{\frac{GM}{r}} \)
- (ii) Maximum distance between the two fragments before collision: \( d = r\sqrt{2} \)
- (iii) Magnitude of their relative velocity just before they collide: \( V_{relative} = \sqrt{\frac{2GM}{r}} \)