Two bodies of masses 4 kg and 5 kg are moving with equal momentum. Then the ratio of their respective kinetic energies is

Options

A. `4:5`
B. `2:1`
C. `1:3`
D. `5:4`

Answer

To solve the problem of finding the ratio of the kinetic energies of two bodies with masses 4 kg and 5 kg that are moving with equal momentum, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between momentum and kinetic energy**:
- The momentum \( p \) of an object is given by the formula:
\[ p = m \cdot v \]
where \( m \) is the mass and \( v \) is the velocity of the object.
- The kinetic energy \( KE \) of an object is given by:
\[ KE = \frac{1}{2} m v^2 \]

2. **Express kinetic energy in terms of momentum**:
- We can express the kinetic energy in terms of momentum. By rearranging the momentum formula, we have:
\[ v = \frac{p}{m} \]
- Substituting this expression for \( v \) into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{1}{2} m \cdot \frac{p^2}{m^2} = \frac{p^2}{2m} \]

3. **Set up the equations for both bodies**:
- Let the first body have mass \( m_1 = 4 \, \text{kg} \) and the second body have mass \( m_2 = 5 \, \text{kg} \).
- Since both bodies have equal momentum \( p \), their kinetic energies can be expressed as:
\[ KE_1 = \frac{p^2}{2m_1} \quad \text{and} \quad KE_2 = \frac{p^2}{2m_2} \]

4. **Calculate the ratio of kinetic energies**:
- The ratio of the kinetic energies \( \frac{KE_1}{KE_2} \) can be calculated as follows:
\[ \frac{KE_1}{KE_2} = \frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{m_2}{m_1} \]
- Substituting the values of \( m_1 \) and \( m_2 \):
\[ \frac{KE_1}{KE_2} = \frac{5 \, \text{kg}}{4 \, \text{kg}} = \frac{5}{4} \]

5. **Conclusion**:
- Therefore, the ratio of the kinetic energies of the two bodies is:
\[ \frac{KE_1}{KE_2} = 5:4 \]