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 find the ratio of the kinetic energies \( E_1 \) and \( E_2 \) of two bodies with different masses \( m_1 \) and \( m_2 \) that have equal momenta, we can follow these steps:

### Step 1: Write the expression for momentum
The momentum \( p \) of an object is given by the formula:
\[ p = m \cdot v \]
For the two bodies, we have:
\[ p_1 = m_1 \cdot v_1 \quad \text{and} \quad p_2 = m_2 \cdot v_2 \]
Since the momenta are equal, we can write:
\[ m_1 \cdot v_1 = m_2 \cdot v_2 \]

### Step 2: Express one velocity in terms of the other
From the equation \( m_1 \cdot v_1 = m_2 \cdot v_2 \), we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = \frac{m_2}{m_1} \cdot v_2 \]

### Step 3: Write the expressions for kinetic energy
The kinetic energy \( E \) of an object is given by:
\[ E = \frac{1}{2} m v^2 \]
Thus, the kinetic energies for the two bodies are:
\[ E_1 = \frac{1}{2} m_1 v_1^2 \quad \text{and} \quad E_2 = \frac{1}{2} m_2 v_2^2 \]

### Step 4: Substitute \( v_1 \) into the kinetic energy expression
Substituting \( v_1 = \frac{m_2}{m_1} \cdot v_2 \) into the expression for \( E_1 \):
\[ E_1 = \frac{1}{2} m_1 \left(\frac{m_2}{m_1} v_2\right)^2 \]
This simplifies to:
\[ E_1 = \frac{1}{2} m_1 \cdot \frac{m_2^2}{m_1^2} \cdot v_2^2 = \frac{1}{2} \cdot \frac{m_2^2}{m_1} \cdot v_2^2 \]

### Step 5: Write the ratio of kinetic energies
Now we can find the ratio \( \frac{E_1}{E_2} \):
\[ \frac{E_1}{E_2} = \frac{\frac{1}{2} \cdot \frac{m_2^2}{m_1} \cdot v_2^2}{\frac{1}{2} m_2 v_2^2} \]
The \( \frac{1}{2} \) and \( v_2^2 \) terms cancel out:
\[ \frac{E_1}{E_2} = \frac{\frac{m_2^2}{m_1}}{m_2} = \frac{m_2}{m_1} \]

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