A cart moves with a constant speed along a horizontal circular path. From the cart, a particle is thrown up vertically with respect to the cart.

Options

A. The particle will land somewhere on the circular path
B. The particle will land outside the circular path
C. The particle will follow an elliptical path
D. The particle will follow a parabolic path

Explanation

To solve the problem of a cart moving with constant speed along a horizontal circular path and a particle being thrown vertically from the cart, we can follow these steps:

### Step 1: Understand the Motion of the Cart
The cart is moving in a circular path with constant speed. This means that while the speed is constant, the direction of the velocity is continuously changing. The velocity of the cart at any point is tangent to the circular path.

**Hint:** Remember that in circular motion, even though speed is constant, the direction of velocity changes, resulting in centripetal acceleration.

### Step 2: Analyze the Particle's Motion
When the particle is thrown vertically upwards from the cart, it has two components of motion:
- A horizontal component (Vx), which is the same as the speed of the cart at the moment of the throw.
- A vertical component (Vy), which is the speed at which the particle is thrown upwards.

**Hint:** Consider the initial velocity of the particle as a vector sum of its horizontal and vertical components.

### Step 3: Determine the Path of the Particle
Since the particle is thrown upwards, it will experience projectile motion. The horizontal motion will be uniform (constant velocity), while the vertical motion will be influenced by gravity, resulting in a parabolic trajectory.

**Hint:** Recall that projectile motion consists of two independent motions: horizontal (constant velocity) and vertical (accelerated motion due to gravity).

### Step 4: Identify the Resultant Path
The resultant path of the particle will be a parabola because it is subject to gravitational acceleration while moving horizontally at the speed of the cart. As the particle moves upward and then downward, it will not follow the circular path of the cart.

**Hint:** Visualize the trajectory of the particle as a parabola opening downwards, starting from the point where it was thrown.

### Step 5: Determine Where the Particle Lands
Since the particle is thrown from a moving cart, it will land outside the circular path traced by the cart. The horizontal distance covered by the particle will depend on the time it takes to reach the ground, which is influenced by the initial vertical velocity and the acceleration due to gravity.

**Hint:** Consider the time of flight and the horizontal distance traveled to conclude where the particle will land relative to the circular path.

### Conclusion
Based on the analysis:
- The path of the particle will be parabolic (Option D).
- The particle will land outside the circular path (Option B).

Thus, the correct answers are Options B and D.

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