The graph between linear monentum (p) and Kinetic energy (k) is best represented by

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the best representation of the graph between linear momentum (p) and kinetic energy (K), we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and momentum
The kinetic energy (K) of an object is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.

The momentum (p) of the object is defined as:
\[ p = mv \]

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

### Step 3: Substitute velocity in the kinetic energy formula
We can substitute this expression for \( v \) back into the kinetic energy formula:
\[ K = \frac{1}{2} m \left(\frac{p}{m}\right)^2 \]
\[ K = \frac{1}{2} m \cdot \frac{p^2}{m^2} \]
\[ K = \frac{p^2}{2m} \]

### Step 4: Identify the relationship between K and p
From the equation \( K = \frac{p^2}{2m} \), we can see that kinetic energy (K) is directly proportional to the square of momentum (p):
\[ K \propto p^2 \]

### Step 5: Determine the graph representation
Since \( K \) is proportional to \( p^2 \), when we plot kinetic energy (K) on the y-axis and momentum (p) on the x-axis, the graph will be a parabola that opens upwards. This is because the relationship is quadratic.

### Step 6: Analyze the options
We need to find the graph that represents this relationship:
- The graph should show a parabola opening upwards.
- At \( p = 0 \), \( K \) should also be \( 0 \).

### Conclusion
Based on the analysis, the graph that best represents the relationship between linear momentum (p) and kinetic energy (K) is the one that shows a parabola opening upwards, which corresponds to option 1.