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

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the correct momentum-height (p-h) diagram for a ball thrown vertically upwards, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- A ball is thrown vertically upwards from the ground with an initial velocity \( v_0 \).
- The momentum \( p \) of the ball at the initial height (ground level, \( h = 0 \)) is given by:
\[ p = m \cdot v_0 \]
- Since the ball is moving upwards, the momentum is positive.

2. **Analyzing Motion**:
- As the ball rises, it experiences a gravitational force acting downwards, which causes its velocity to decrease.
- At the maximum height \( h \), the velocity becomes zero, and thus the momentum also becomes zero:
\[ p = 0 \quad \text{at} \quad h = h_{\text{max}} \]

3. **Momentum as a Function of Height**:
- The relationship between momentum and height can be derived from the equations of motion. Using the equation:
\[ v^2 = u^2 - 2gh \]
- Here, \( v \) is the final velocity (which becomes zero at maximum height), \( u \) is the initial velocity \( v_0 \), and \( g \) is the acceleration due to gravity.
- Rearranging gives:
\[ 0 = v_0^2 - 2gh_{\text{max}} \implies h_{\text{max}} = \frac{v_0^2}{2g} \]

4. **Momentum-Height Relationship**:
- The momentum can be expressed as:
\[ p = m \cdot v = m \cdot \sqrt{v_0^2 - 2gh} \]
- Squaring both sides gives:
\[ p^2 = m^2(v_0^2 - 2gh) \]
- This shows that \( p^2 \) is a linear function of \( h \), indicating that the momentum decreases as height increases.

5. **Graphing the Relationship**:
- At \( h = 0 \), \( p \) is maximum (\( p = m v_0 \)).
- As \( h \) increases to \( h_{\text{max}} \), \( p \) decreases to zero.
- After reaching maximum height, as the ball falls back down, momentum becomes negative, increasing in magnitude until it reaches \( -m v_0 \) when it returns to the ground.

6. **Identifying the Correct Option**:
- The correct momentum-height graph will show momentum starting at a maximum positive value at \( h = 0 \), decreasing to zero at maximum height, and then becoming negative as the ball falls back down.
- This behavior corresponds to option D.

### Final Answer:
The correct momentum-height (p-h) diagram is **option D**.