If K, p and m represent the kinetic energy, momentum and mass of a body respectively, then which of the following graphs is incorrect?

Options

A.
B.
C.
D.

Answer

To determine which graph is incorrect among the relationships of kinetic energy (K), momentum (P), and mass (m), we will analyze the relationships mathematically and graphically.

### Step 1: Understand the Relationships
1. **Kinetic Energy (K)**: The formula for kinetic energy is given by:
\[ K = \frac{1}{2} mv^2 \]
2. **Momentum (P)**: The formula for momentum is given by:
\[ P = mv \]

### Step 2: Analyze Each Graph
#### Graph 1: K vs. M (Keeping P Constant)
- If we keep momentum (P) constant, we can express velocity (v) in terms of momentum:
\[ v = \frac{P}{m} \]
- Substituting this into the kinetic energy formula gives:
\[ K = \frac{1}{2} m \left(\frac{P}{m}\right)^2 = \frac{P^2}{2m} \]
- This shows that kinetic energy (K) is inversely proportional to mass (m). Therefore, the graph of K vs. M will be a hyperbola, which is correct.

#### Graph 2: K vs. P (Keeping M Constant)
- The relationship can be derived from the kinetic energy formula:
\[ K = \frac{P^2}{2m} \]
- This indicates that K is directly proportional to \( P^2 \). Therefore, the graph of K vs. P will be a quadratic curve, which is correct.

#### Graph 3: Root K vs. P (Keeping M Constant)
- If we take the square root of the kinetic energy:
\[ \sqrt{K} = \sqrt{\frac{P^2}{2m}} = \frac{P}{\sqrt{2m}} \]
- This shows that \(\sqrt{K}\) is directly proportional to P. Hence, the graph of \(\sqrt{K}\) vs. P will be a straight line, which is correct.

#### Graph 4: P vs. M (Keeping K Constant)
- If we keep kinetic energy (K) constant, we can rearrange the kinetic energy formula:
\[ K = \frac{P^2}{2m} \implies P^2 = 2Km \implies P = \sqrt{2Km} \]
- This indicates that P is proportional to the square root of m. Therefore, the graph of P vs. M will be a curve, not a straight line.

### Step 3: Identify the Incorrect Graph
From our analysis:
- Graph 1: Correct (K vs. M is a hyperbola)
- Graph 2: Correct (K vs. P is a quadratic curve)
- Graph 3: Correct (\(\sqrt{K}\) vs. P is a straight line)
- **Graph 4: Incorrect (P vs. M should be a curve, not a straight line)**

### Conclusion
The incorrect graph is **Graph 4**, which depicts the relationship between momentum (P) and mass (m) while keeping kinetic energy (K) constant.

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