QJEEM24Ph5a98: JEE Main 2024 Physics - Class 11 JEEM24Ph5a98 Solution: A stationary particle breaks into two parts of masses and which …

A stationary particle breaks into two parts with masses $m_A$ and $m_B$ , which then move with velocities $v_A$ and $v_B$ , respectively. We need to determine the ratio of their kinetic energies $K_A$ and $K_B$ .

Since the initial momentum of the particle is zero, the momentum of the two parts must be equal and opposite to conserve momentum:

$m_A v_A = m_B v_B$

Here, the ratio of kinetic energies is given by:

$\frac{K_A}{K_B} = \frac{\frac{1}{2} m_A v_A^2}{\frac{1}{2} m_B v_B^2} = \frac{m_A v_A^2}{m_B v_B^2}$

However, using the momentum relationship, we can substitute $m_A v_A = m_B v_B$ into the kinetic energy ratio, leading us to simplify:

$\frac{K_A}{K_B} = \frac{v_A}{v_B}$

Therefore, the ratio of their kinetic energies $\left(K_B: K_A\right)$ is:

$\frac{K_B}{K_A} = \frac{v_B}{v_A}$