A ball is thrown up with a certain velocity at anangle `theta` to the horizontal. The kinetic energy KE of the ball varies with horizontal displacements as:

Options

A.
B.
C.
D.

Explanation

To solve the problem of how the kinetic energy (KE) of a ball thrown at an angle `theta` to the horizontal varies with horizontal displacement `s`, we can analyze the motion of the projectile step by step.

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
- A ball thrown at an angle `theta` has both horizontal and vertical components of velocity.
- The initial velocity can be broken down into:
- Horizontal component: \( V_{x} = V_0 \cos(\theta) \)
- Vertical component: \( V_{y} = V_0 \sin(\theta) \)

2. **Kinetic Energy Formula**:
- The kinetic energy of the ball is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
- Where \( v \) is the magnitude of the velocity of the ball.

3. **Velocity at Different Points**:
- As the ball rises, the vertical component of the velocity \( V_{y} \) decreases due to gravity until it reaches the highest point (Hmax).
- At the highest point, the vertical component \( V_{y} \) becomes zero, and only the horizontal component \( V_{x} \) remains:
\[ V = V_{x} = V_0 \cos(\theta) \]
- As the ball descends, the vertical component \( V_{y} \) increases again.

4. **Kinetic Energy Variation**:
- As the ball moves upwards, the vertical component of the velocity decreases, leading to a decrease in kinetic energy.
- When the ball reaches the highest point, the kinetic energy is:
\[ KE_{max} = \frac{1}{2} m (V_0 \cos(\theta))^2 \]
- After reaching the highest point, as the ball falls, the vertical component of the velocity increases, thus increasing the total kinetic energy again.

5. **Graph of Kinetic Energy vs. Horizontal Displacement**:
- From the analysis, we can conclude:
- From the launch to the highest point, the kinetic energy decreases.
- From the highest point to the landing point, the kinetic energy increases.
- Therefore, the kinetic energy varies in a parabolic manner, reaching a minimum at the highest point.

6. **Conclusion**:
- The correct option is that the kinetic energy decreases to a minimum at the highest point and then increases again, which corresponds to the third option provided in the question.