A particle reaches its highest point when it has covered exactly one half of its horizontal range. The corresponding point on the displacement -time graph is charecterized by :

Options

A. negative slope and zero curvature
B. zero slope and negative curvature
C. zero slope and positive curvature
D. positive slope and zero curvature

Explanation

To solve the problem, we need to analyze the motion of a particle projected at an angle and determine the characteristics of the displacement-time graph when the particle reaches its highest point after covering half of its horizontal range.

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
- When a particle is projected at an angle θ, it follows a parabolic trajectory due to the influence of gravity. The horizontal range (R) is the total horizontal distance covered when the particle returns to the same vertical level from which it was projected.

2. **Identifying the Highest Point**:
- The highest point of the projectile's path is reached when the vertical component of its velocity becomes zero. At this point, the particle has covered half of its horizontal range (R/2).

3. **Analyzing the Displacement-Time Graph**:
- The displacement in the vertical direction (y) will be at its maximum at the highest point, while the horizontal displacement (x) will be at R/2.
- On a displacement-time graph, the y-axis represents displacement and the x-axis represents time.

4. **Slope of the Displacement-Time Graph**:
- The slope of the displacement-time graph at any point represents the velocity of the particle at that time.
- At the highest point, the vertical velocity is zero (the particle is momentarily at rest in the vertical direction), which means the slope of the graph (dy/dt) at this point is zero.

5. **Curvature of the Displacement-Time Graph**:
- The curvature of the graph indicates how the slope is changing.
- Since the particle is transitioning from moving upwards to moving downwards, the graph will have a positive curvature at the highest point. This is because the graph will be concave upwards at this point.

6. **Conclusion**:
- Therefore, at the highest point where the particle has covered half of its horizontal range, the displacement-time graph is characterized by:
- A slope of zero (indicating zero vertical velocity).
- Positive curvature (indicating the transition from increasing to decreasing displacement).

### Final Answer:
The corresponding point on the displacement-time graph is characterized by a slope of zero and positive curvature.