A particle of mass m is revolving around a planet in a circular orbit of radius R. At the instant the particle has velocity `vec V` another particle of mass `m/2` moving at velocity ` (vec V)/2` collides perfectly in-elastically with the first particle. The new path of the combined body will take is

Options

A. continues to move in a circular orbit
B. starts moving in a elliptical orbit around the planet
C. Escapes from the planet's Gravitational field
D. Falls vertically downwards towards the planet

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Conditions
We have two particles:
- Particle 1 (mass = m) is revolving around a planet in a circular orbit with velocity \( \vec{V} \).
- Particle 2 (mass = \( \frac{m}{2} \)) is moving with velocity \( \frac{\vec{V}}{2} \).

### Step 2: Apply Conservation of Momentum
Since the collision is perfectly inelastic, we can use the conservation of momentum. The total momentum before the collision must equal the total momentum after the collision.

The momentum before the collision:
\[ \text{Momentum}_{\text{initial}} = m \vec{V} + \frac{m}{2} \left(\frac{\vec{V}}{2}\right) = m \vec{V} + \frac{m}{4} \vec{V} = \left(m + \frac{m}{4}\right) \vec{V} = \frac{5m}{4} \vec{V} \]

Let \( \vec{V}' \) be the velocity of the combined mass after the collision. The total mass after the collision is \( m + \frac{m}{2} = \frac{3m}{2} \).

The momentum after the collision:
\[ \text{Momentum}_{\text{final}} = \frac{3m}{2} \vec{V}' \]

Setting the initial momentum equal to the final momentum:
\[ \frac{5m}{4} \vec{V} = \frac{3m}{2} \vec{V}' \]

### Step 3: Solve for the New Velocity \( \vec{V}' \)
To find \( \vec{V}' \), we can simplify the equation:
\[ \frac{5}{4} \vec{V} = \frac{3}{2} \vec{V}' \]

Rearranging gives:
\[ \vec{V}' = \frac{5}{4} \cdot \frac{2}{3} \vec{V} = \frac{5}{6} \vec{V} \]

### Step 4: Analyze the New Velocity
The new velocity \( \vec{V}' = \frac{5}{6} \vec{V} \) is less than the original orbital velocity \( \vec{V} \).

### Step 5: Determine the New Path
Since the new velocity is less than the required orbital velocity, the combined body will no longer maintain a circular orbit. Instead, it will move towards the center of the planet due to gravitational attraction.

In orbital mechanics, if an object’s velocity is less than the circular orbital velocity, it will follow an elliptical path as it is pulled inward by gravity.

### Conclusion
Therefore, the new path of the combined body will be an **elliptical orbit** around the planet.

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