A particle is projected from a point of an angle with the horizontal. At any instant t, if p is the linear momentum and E the kinetic energy, then which of the following graph is/are correct?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationship between linear momentum (p) and kinetic energy (E) of a particle projected at an angle with the horizontal.

### Step-by-Step Solution:

1. **Understanding Linear Momentum and Kinetic Energy**:
- The linear momentum (p) of a particle is given by the formula:
\[ p = mv \]
where \(m\) is the mass of the particle and \(v\) is its velocity.
- The kinetic energy (E) of the particle is given by:
\[ E = \frac{1}{2} mv^2 \]

2. **Relating Kinetic Energy to Linear Momentum**:
- We can express kinetic energy in terms of momentum. Since \(p = mv\), we can solve for \(v\):
\[ v = \frac{p}{m} \]
- Substituting this expression for \(v\) into the kinetic energy formula:
\[ E = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m} \]
- This shows that kinetic energy (E) is directly proportional to the square of the momentum (p).

3. **Graphical Representation**:
- From the relationship \(E = \frac{p^2}{2m}\), we can deduce the nature of the graph:
- If we plot \(E\) against \(p^2\), the equation resembles \(y = kx\) (a straight line) where \(k = \frac{1}{2m}\).
- If we plot \(E\) against \(p\), the equation resembles \(y = kx^2\) (a parabola) because \(E\) is proportional to \(p^2\).

4. **Analyzing the Options**:
- **Option A**: Graph between E and P (should be a parabola) - **False**.
- **Option B**: Graph between E and P^2 (should be a straight line) - **True**.
- **Option C**: Graph between E and P (should be an upward parabola) - **False**.
- **Option D**: Graph between E and P^2 (should be a straight line) - **True**.

5. **Conclusion**:
- The correct graphs are those that show the relationship between kinetic energy and the square of momentum, which is a straight line through the origin. Therefore, the correct answer is the graph of E vs P^2.

### Final Answer:
The correct graph is the one that represents the relationship between kinetic energy (E) and the square of momentum (P^2), which is a straight line.