If p and E represent linear momentum and kinetic energy respectively, then the graph between E and pis correctly shown by

Options

A.
B.
C.
D.

Explanation

To find the correct graph between linear momentum (p) and kinetic energy (E), we start with the definitions of kinetic energy and momentum.

### Step 1: Write the formulas for kinetic energy and momentum.
- Kinetic Energy (E) is given by the formula:
\[ E = \frac{1}{2} mv^2 \]
- Momentum (p) is given by the formula:
\[ p = mv \]

### Step 2: Express velocity (v) in terms of momentum (p).
From the momentum equation, we can express velocity as:
\[ v = \frac{p}{m} \]

### Step 3: Substitute the expression for velocity into the kinetic energy formula.
Now, we substitute \(v\) into the kinetic energy equation:
\[ E = \frac{1}{2} m \left(\frac{p}{m}\right)^2 \]
This simplifies to:
\[ E = \frac{1}{2} m \cdot \frac{p^2}{m^2} = \frac{p^2}{2m} \]

### Step 4: Analyze the relationship between E and p.
From the equation \(E = \frac{p^2}{2m}\), we can see that kinetic energy (E) is directly proportional to the square of momentum (p):
\[ E \propto p^2 \]
This indicates that if we plot E against p, the graph will be a parabola opening upwards.

### Step 5: Determine the characteristics of the graph.
- The graph will pass through the origin (0,0) because when \(p = 0\), \(E\) will also be \(0\).
- The shape of the graph is a parabola because \(E\) is proportional to \(p^2\).

### Step 6: Identify the correct option from the given choices.
Given the characteristics of the graph, we look for a graph that is a parabola opening upwards and passes through the origin.

### Conclusion:
The correct graph between kinetic energy (E) and linear momentum (p) is a parabola that opens upwards, which corresponds to **Option 2**.

---