Which of the following graphs represents the graphical relation between momentum (p) and kinetic energy (K) for a body in motion?

Options

A.
B.
C.
D. None

Answer

To determine the graphical relation between momentum (p) and kinetic energy (K) for a body in motion, we can follow these steps:

### Step-by-Step Solution:

1. **Define Momentum and Kinetic Energy**:
- Momentum (p) is defined as the product of mass (m) and velocity (v):
\[ p = mv \]
- Kinetic Energy (K) is defined as:
\[ K = \frac{1}{2} mv^2 \]

2. **Express Kinetic Energy in terms of Momentum**:
- We can express \(K\) in terms of \(p\) by substituting \(v\) from the momentum equation:
\[ v = \frac{p}{m} \]
- Substituting this into the kinetic energy equation:
\[ K = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{1}{2} m \frac{p^2}{m^2} = \frac{p^2}{2m} \]

3. **Analyze the Relationship**:
- The equation \(K = \frac{p^2}{2m}\) shows that kinetic energy \(K\) is proportional to the square of momentum \(p\). This means that if we plot \(K\) against \(p^2\), we will get a linear relationship.

4. **Graphical Representation**:
- If we plot \(K\) on the y-axis and \(p^2\) on the x-axis, the graph will be a straight line with a slope of \(\frac{1}{2m}\).
- However, if we plot \(K\) against \(p\), the relationship is quadratic, and the graph will be a parabola opening upwards.

5. **Conclusion**:
- The correct representation of the relationship between momentum and kinetic energy is not a straight line through the origin, but rather a parabolic curve when plotted as \(K\) versus \(p\). Therefore, the answer to the question is that none of the provided options accurately represent this relationship.