A ball is thrown upwards and returns to the ground describing a parabolic path. Which of the following quantities remains constant ?

Options

A. Kinetic energy of the ball
B. The speed of the ball
C. The vertical component of velocity
D. The horizontal component of velocity

Explanation

To solve the question regarding which quantities remain constant when a ball is thrown upwards and returns to the ground describing a parabolic path, we can analyze the motion step by step.

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
- When a ball is thrown upwards, it follows a parabolic trajectory due to the influence of gravity. The motion can be broken down into horizontal and vertical components.

2. **Analyzing Horizontal Motion**:
- In the horizontal direction (x-axis), there is no acceleration acting on the ball (assuming air resistance is negligible). Therefore, the horizontal component of velocity (u_x) remains constant throughout the motion.

3. **Analyzing Vertical Motion**:
- In the vertical direction (y-axis), the ball is subject to gravitational acceleration (g). This means that the vertical component of velocity (u_y) changes over time. Initially, it is positive (upwards) and decreases to zero at the peak of the trajectory, then becomes negative (downwards) as it falls back to the ground.

4. **Kinetic Energy**:
- The kinetic energy (K.E.) of the ball is given by the formula \( K.E. = \frac{1}{2} m v^2 \), where \( v \) is the magnitude of the velocity. Since both the horizontal and vertical components of velocity change, the total speed of the ball (which is the vector sum of the horizontal and vertical components) will also change. Therefore, the kinetic energy is not constant.

5. **Conclusion**:
- The only quantity that remains constant during the entire motion is the horizontal component of velocity (u_x). The vertical component of velocity (u_y) and the total speed of the ball change throughout the motion, and the kinetic energy also varies.

### Final Answer:
The quantity that remains constant is the **horizontal component of velocity (u_x)**.

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