Question

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

• A. 4:5
• B. 2:1
• C. 1:3
• D. 5:4

Hint:

When two objects have equal momentum, the ratio of their kinetic energies is the inverse of the ratio of their masses.

Answer 1

The Correct Option is D

The kinetic energy (\( K.E. \)) of a body is given by: \[ K.E. = \frac{1}{2} m v^2 \] where \( m \) is the mass and \( v \) is the velocity of the object. Momentum (\( p \)) is given by: \[ p = mv \] Since the momentum of both bodies is equal, we have: \[ p_1 = p_2 \] \[ m_1 v_1 = m_2 v_2 \] Substitute \( v_1 \) and \( v_2 \) in terms of momentum: \[ v_1 = \frac{p}{m_1}, \quad v_2 = \frac{p}{m_2} \] Now, calculate the kinetic energy for both bodies: \[ K.E_1 = \frac{1}{2} m_1 \left(\frac{p}{m_1}\right)^2 = \frac{p^2}{2 m_1} \] \[ K.E_2 = \frac{1}{2} m_2 \left(\frac{p}{m_2}\right)^2 = \frac{p^2}{2 m_2} \] Thus, the ratio of their kinetic energies is: \[ \frac{K.E_1}{K.E_2} = \frac{\frac{p^2}{2 m_1}}{\frac{p^2}{2 m_2}} = \frac{m_2}{m_1} = \frac{5}{4} \]