A particle reaches its highest point when it has covered exactly one-half of its horizontal range. The corresponding point on the vertical displacement time graph is characterized by:
A. Zero slope and zero curvature
B. Zero slope and non-zero curvature
C. Positive slope and zero curvature
D. None of these

Hint:On a displacement-time graph, a sloping line indicates that the item is moving. In a displacement-time curve, the line's slope or gradient is equal to the object's velocity. The quicker the item moves, the larger the gradient and the steeper the line.

Complete step-by-step answer:
We can say that a vertical displacement time graph will have the same trajectory as that of the particle. We can say that when the particle has covered half the distance of its horizontal range, or when it is at the highest point of the trajectory, it will have a non-zero curvature as the point itself does not define whether the trajectory is a straight line or is it in the form of a curve. So, by joining the points before and after the highest point followed by the particle, we can say that it has non-zero curvature.

At the initial stage when the particle starts to move in the upward direction, we can say that its slope will be positive and highest at that moment, and gradually the slope will become steeper and then eventually the slope will be negative before it reaches the ground again. So, we can say that at the highest point the slope of the particle will be steepest, that is its slope will be zero.

So, option B, zero slope and non-zero curvature is the required solution.

Note: When finding the curvature and the slope of the particle, try to imagine it as a vertical parabola to make its visualization more clear as to how the trajectory of the particle will be.