Explanation:

When a particle is thrown up vertically with respect to the moving cart, there are two factors to consider: the initial velocity of the particle and the circular motion of the cart.

1. Initial Velocity of the Particle:
When the particle is thrown up, it has an initial velocity with respect to the cart. This initial velocity can have three possibilities:
- It can be equal to the linear velocity of the cart.
- It can be greater than the linear velocity of the cart.
- It can be less than the linear velocity of the cart.

2. Circular Motion of the Cart:
The cart is moving along a horizontal circular path with a constant speed. This circular motion can also have three possibilities:
- The radius of the circular path can be larger than the distance covered by the particle vertically.
- The radius of the circular path can be equal to the distance covered by the particle vertically.
- The radius of the circular path can be smaller than the distance covered by the particle vertically.

Combining the Two Factors:
Now, let's consider the combinations of the initial velocity of the particle and the circular motion of the cart:

1. Initial Velocity equal to Linear Velocity of the Cart:
If the initial velocity of the particle is equal to the linear velocity of the cart, then the particle will land on the circular path. This is because the particle will continue to move horizontally along with the cart while also moving vertically due to its initial velocity.

2. Initial Velocity greater than Linear Velocity of the Cart:
If the initial velocity of the particle is greater than the linear velocity of the cart, then the particle will land outside the circular path. This is because the particle has a greater initial velocity, and as it moves vertically, it will cover a greater distance compared to the distance covered by the cart horizontally.

3. Initial Velocity less than Linear Velocity of the Cart:
If the initial velocity of the particle is less than the linear velocity of the cart, then the particle can follow two possible paths:
- If the radius of the circular path is larger than the distance covered by the particle vertically, then the particle will land on the circular path. This is because the particle will not be able to cover the entire circular path vertically and will eventually come back down to the same horizontal position as the cart.
- If the radius of the circular path is smaller than the distance covered by the particle vertically, then the particle will follow a parabolic path. This is because the particle has a smaller radius to cover in the horizontal direction compared to the distance covered vertically, resulting in a parabolic trajectory.

Conclusion:
Based on the combinations of the initial velocity of the particle and the circular motion of the cart, the correct options are:
- The particle will land somewhere on the circular path (Option B).
- The particle will follow a parabolic path (Option D).