Two bodies of different masses `m_(1)` and `m_(2)` have equal momenta. Their kinetic energies `E_(1)` and `E_(2)` are in the ratio

Options

A. `sqrt((m_(2))/(m_(1)))`
B. `sqrt((m_(1))/(m_(2)))`
C. `(m_(1))/(m_(2))`
D. `(m_(2))/(m_(1))`

Answer

To solve the problem, we need to establish the relationship between the kinetic energies of two bodies with equal momentum. Let's denote the two bodies as Body 1 with mass \( m_1 \) and Body 2 with mass \( m_2 \).

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

The kinetic energy \( E \) of an object is given by the formula:
\[ E = \frac{1}{2} mv^2 \]

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

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

### Step 4: Write the kinetic energies for both bodies
For Body 1:
\[ E_1 = \frac{p^2}{2m_1} \]
For Body 2:
\[ E_2 = \frac{p^2}{2m_2} \]

### Step 5: Find the ratio of the kinetic energies
Now, we can find the ratio of the kinetic energies \( E_1 \) and \( E_2 \):
\[ \frac{E_1}{E_2} = \frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{m_2}{m_1} \]

### Step 6: Conclusion
Thus, the ratio of the kinetic energies \( E_1 \) and \( E_2 \) is:
\[ \frac{E_1}{E_2} = \frac{m_2}{m_1} \]
This implies that the kinetic energies are inversely proportional to their respective masses when the momentum is constant.

### Final Answer
The ratio of the kinetic energies \( E_1 \) and \( E_2 \) is:
\[ \frac{E_1}{E_2} = \frac{m_2}{m_1} \]