91Ó°ÊÓ

First, we need to calculate the initial kinetic energy of each mass. The formula for kinetic energy is: \[KE = \frac{1}{2}mv^2\] For mass 1 (1 kg), moving with a velocity of 3 m/s: \[KE_1 = \frac{1}{2}(1)(3^2) = \frac{1}{2}(9) = 4.5 J\] For mass 2 (2 kg), moving with a velocity of 2 m/s: \[KE_2 = \frac{1}{2}(2)(2^2) = (1)(4) = 4 J\]

Now, we need to add the initial kinetic energy of both masses to find the total initial kinetic energy: \[KE_{total} = KE_1 + KE_2 = 4.5 J + 4 J = 8.5 J\]

Since both bodies stick together after collision, we can use the conservation of momentum to find the final velocity of the combined mass. The equation for conservation of momentum is: \[m_1 \vec{v_1} + m_2 \vec{v_2} = (m_1 + m_2) \vec{v_f}\] Since both masses are moving in mutually perpendicular directions, we can treat this as a vector sum: \[\vec{v_f} = \frac{\vec{v_1} + \vec{v_2}}{m_1 + m_2}\] \[\vec{v_f} = \frac{1 \cdot \vec{3} + 2 \cdot \vec{2}}{1+2}\] Here, one of the velocities is in one direction (e.g., x-axis) and the other velocity is in another direction (e.g., y-axis). So we'll get the final velocity in the form of (a, b), where a and b are the components of the final velocity: \[\vec{v_f}= \left(\frac{1 \cdot 3}{3}, \frac{2 \cdot 2}{3}\right)=\left(1, \frac{4}{3}\right)\] The magnitude of the final velocity is: \[|\vec{v_f}|=\sqrt{1^2+\left(\frac{4}{3}\right)^2}=\sqrt{1+\frac{16}{9}}=\sqrt{\frac{25}{9}}=\frac{5}{3}\mathrm{~m/s}\]

Now, we need to find the final kinetic energy of the combined mass, using the formula for kinetic energy: \[KE_f = \frac{1}{2}(m_1+m_2)(v_f)^2\] \[KE_f = \frac{1}{2}(3)\left(\frac{5}{3}\right)^2 = \frac{1}{2}(3)\left(\frac{25}{9}\right) = \frac{25}{6} J\]

Lastly, to calculate the energy loss, subtract the final kinetic energy from the total initial kinetic energy: \[Energy\:Loss = KE_{total} - KE_f = 8.5J - \frac{25}{6} J = \frac{17}{2} J - \frac{25}{6} J =\frac{13}{6} J\] Thus, the energy loss is \(\frac{13}{6} J\), which is not one of the given options. However, it is close to option (B) \(\frac{13}{3} \mathrm{~J}\).