Two bodies of different masses are moving with same kinetic energy. Then, the ratio of their moment is equal to the ratio of their

Options

A. masses
B. square of masses
C. square root of masses
D. cube root of masses

Explanation

To solve the problem, we need to find the ratio of the momenta of two bodies that have the same kinetic energy. Let's denote the two bodies as Body 1 and Body 2, with masses \( m_1 \) and \( m_2 \), and their respective momenta as \( p_1 \) and \( p_2 \).

### Step 1: Write the formula for kinetic energy
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the object.

### Step 2: Set up the kinetic energy equations for both bodies
Since both bodies have the same kinetic energy, we can write:
\[ KE_1 = KE_2 \]
This gives us:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]
We can simplify this by canceling out \(\frac{1}{2}\):
\[ m_1 v_1^2 = m_2 v_2^2 \]

### Step 3: Express momentum in terms of mass and velocity
The momentum \( p \) of an object is given by:
\[ p = mv \]
Thus, for the two bodies, we have:
\[ p_1 = m_1 v_1 \quad \text{and} \quad p_2 = m_2 v_2 \]

### Step 4: Relate momentum to kinetic energy
From the kinetic energy equation, we can express \( v_1^2 \) and \( v_2^2 \) in terms of momentum:
\[ v_1 = \frac{p_1}{m_1} \quad \text{and} \quad v_2 = \frac{p_2}{m_2} \]
Substituting these into the kinetic energy equation gives:
\[ m_1 \left(\frac{p_1}{m_1}\right)^2 = m_2 \left(\frac{p_2}{m_2}\right)^2 \]
This simplifies to:
\[ \frac{p_1^2}{m_1} = \frac{p_2^2}{m_2} \]

### Step 5: Find the ratio of momenta
Rearranging the above equation gives:
\[ \frac{p_1^2}{p_2^2} = \frac{m_1}{m_2} \]
Taking the square root of both sides, we find:
\[ \frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}} \]

### Conclusion
Thus, the ratio of their momenta \( p_1 \) and \( p_2 \) is equal to the square root of the ratio of their masses \( m_1 \) and \( m_2 \).

### Final Answer
The ratio of their momenta is equal to the ratio of the square root of their masses.