Two unequal masses possess the same K.E. Then, the heavier mass has

Options

A. greater momentum
B. smaller momentum
C. the same momentum as the lighter mass
D. greater speed

Answer

To solve the problem of comparing the momentum of two unequal masses that possess the same kinetic energy, we can follow these steps:

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

The momentum (P) of an object is given by the formula:
\[ P = mv \]

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

### Step 3: Substitute velocity into the kinetic energy formula
Substituting \( v \) into the kinetic energy formula gives:
\[ K.E. = \frac{1}{2} m \left(\frac{P}{m}\right)^2 \]
This simplifies to:
\[ K.E. = \frac{P^2}{2m} \]

### Step 4: Set up the equations for both masses
Let’s denote the two masses as \( m_1 \) (heavier mass) and \( m_2 \) (lighter mass). Since both have the same kinetic energy \( K \), we can write:
\[ K = \frac{P_1^2}{2m_1} \]
\[ K = \frac{P_2^2}{2m_2} \]

### Step 5: Equate the two expressions for kinetic energy
Since both expressions equal \( K \), we can set them equal to each other:
\[ \frac{P_1^2}{2m_1} = \frac{P_2^2}{2m_2} \]

### Step 6: Rearrange to find the relationship between momenta
By cross-multiplying, we get:
\[ P_1^2 m_2 = P_2^2 m_1 \]
This implies:
\[ \frac{P_1^2}{P_2^2} = \frac{m_1}{m_2} \]

### Step 7: Analyze the implications of the masses
Since \( m_1 > m_2 \) (the heavier mass is \( m_1 \)), it follows that:
\[ \frac{P_1^2}{P_2^2} > 1 \]
This means:
\[ P_1 > P_2 \]
Thus, the heavier mass \( m_1 \) has greater momentum than the lighter mass \( m_2 \).

### Conclusion
The heavier mass has greater momentum.