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. Zero slope and zero curvature
B. Zero slope and non-zero curvature
C. Positive slope and zero curvature
D. None of these

Explanation

To solve the problem, we need to analyze the motion of a projectile and how it relates to the displacement-time graph at the highest point of its trajectory. Let's break it down step by step:

### Step 1: Understand the Motion of the Particle
The problem states that a particle reaches its highest point when it has covered exactly one-half of its horizontal range. This means that at the highest point, the vertical component of its velocity becomes zero, while the horizontal component remains constant.

### Step 2: Analyze the Displacement-Time Graph
The displacement-time graph represents how the displacement of the particle changes over time. The slope of this graph at any point gives us the velocity of the particle at that instant.

### Step 3: Identify the Highest Point
At the highest point of the projectile's motion:
- The vertical component of the velocity is zero (the particle is not moving up or down).
- The horizontal component of the velocity (u cos θ) is still present, meaning the particle is still moving forward.

### Step 4: Determine the Slope at the Highest Point
Since the vertical velocity is zero, the only velocity contributing to the slope of the displacement-time graph is the horizontal component. Therefore, the slope of the displacement-time graph at the highest point is positive, indicating that the particle is still moving in the positive x-direction.

### Step 5: Assess the Curvature of the Graph
The curvature of the displacement-time graph indicates how the velocity is changing over time. At the highest point, the velocity is constant (the horizontal component is not changing), which means there is no curvature at this point.

### Conclusion
Based on the analysis:
- The slope of the displacement-time graph at the highest point is positive (since the particle is still moving forward).
- The curvature of the graph is zero (since the velocity is constant).

Thus, the correct characterization of the corresponding point on the displacement-time graph is: **Positive slope and zero curvature.**