A ball is thrown vertically up (taken as +z-axis) from the ground. The correct momentum-height (p-h) diagram is:

Problem Statement

A ball is thrown vertically up (taken as $$+z\text{-axis}$$) from the ground. The correct momentum-height ($$p\text{-}h$$) diagram is:

• (1) Diagram 1
• (2) Diagram 2
• (3) Diagram 3
• (4) Diagram 4
• Initial velocity = $$u$$ (positive, upwards)
• Acceleration due to gravity = $$-g$$ (downwards)
• Displacement $$s$$ = height $$h$$
• The ball starts at the ground ($$h = 0$$) with maximum upward velocity ($$+u$$). Thus, momentum is at its maximum positive value ($$+p_{\text{max}}$$).
• As height $$h$$ increases, the ball slows down until it reaches maximum height ($$h_{\text{max}}$$), where velocity and momentum become zero ($$p = 0$$).
• Trajectory: The path moves from the positive $$p\text{-axis}$$ downward and rightward toward the $$h\text{-axis}$$ (indicated by an arrow pointing down and right).
• From maximum height ($$h_{\text{max}}$$), the ball begins falling back down. Its height decreases back toward zero.
• Since the motion is downward (opposite to $$+z$$), the velocity and momentum become increasingly negative (moving into the $$-p$$ region).
• Trajectory: The path moves from the $$h\text{-axis}$$ downward and leftward toward the negative $$p\text{-axis}$$ (indicated by an arrow pointing down and left).
• Option (3) correctly shows the arrows continuously pointing along the chronological flow of time (first moving towards maximum height, then returning downwards with negative momentum).

Step-by-Step Derivation

To find the correct relation between momentum ($$p$$) and height ($$h$$), we use kinematic equations.

1. Mathematical Relationship

From the third equation of motion:

$$v^2 = u^2 + 2as$$

Given parameters for a ball thrown vertically upward:

Substituting these gives:

$$v^2 = u^2 - 2gh$$

Since momentum is defined as $$p = mv \implies v = \frac{p}{m}$$, we substitute this into the equation:

$$\left(\frac{p}{m}\right)^2 = u^2 - 2gh$$

$$p^2 = m^2(u^2 - 2gh)$$

$$p^2 = m^2u^2 - 2m^2gh$$

This equation is of the form $$p^2 = \text{constant} - kh$$, which mathematically represents a parabola symmetric about the $$h\text{-axis}$$ and opening to the left. This eliminates options (1) and (2) which are straight lines.

2. Direction Analysis (Sign of Momentum)

Let's break down the motion into two halves to determine the correct direction arrows:

First Half: Upward Journey

Second Half: Downward Journey

Conclusion

Combining both halves results in a smooth parabolic curve starting on the positive vertical axis, looping down through $$h_{\text{max}}$$, and finishing on the negative vertical axis.

Looking closely at the directional arrows:

Therefore, the correct option is (3).