Two unequal masses possess the same momen- tum, then the kinetic energy of the heavier mass Is ___ the kinetic energy of the lighter mass.

Options

A. same as
B. greater than
C. smaller than
D. much greater than

Answer

To solve the question, "Two unequal masses possess the same momentum, then the kinetic energy of the heavier mass is ___ the kinetic energy of the lighter mass," we can follow these steps:

### Step 1: Understand the relationship between momentum and kinetic energy
Momentum (p) is defined as the product of mass (m) and velocity (v):
\[ p = m \cdot v \]

Kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]

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

### Step 3: Substitute velocity into the kinetic energy formula
Now, we can substitute this expression for velocity into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2 \]
\[ KE = \frac{1}{2} m \cdot \frac{p^2}{m^2} \]
\[ KE = \frac{p^2}{2m} \]

### Step 4: Write the kinetic energy for both masses
Let’s denote the heavier mass as \( m_1 \) and the lighter mass as \( m_2 \) (where \( m_1 > m_2 \)). Since both masses have the same momentum \( p \):
- Kinetic energy of the heavier mass:
\[ KE_1 = \frac{p^2}{2m_1} \]
- Kinetic energy of the lighter mass:
\[ KE_2 = \frac{p^2}{2m_2} \]

### Step 5: Compare the kinetic energies
Since \( m_1 > m_2 \), we can compare the kinetic energies:
- Because \( m_1 \) is greater than \( m_2 \), the fraction \( \frac{1}{m_1} \) will be less than \( \frac{1}{m_2} \):
\[ KE_1 < KE_2 \]

### Conclusion
Thus, we conclude that the kinetic energy of the heavier mass is **less than** the kinetic energy of the lighter mass.

### Final Answer
The kinetic energy of the heavier mass is **less than** the kinetic energy of the lighter mass.

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