Step 1: Calculate the total initial momentum. The initial momentum of the system is the sum of the individual momenta of the two objects. $$P_{\text{initial}} = m_1 u_1 + m_2 u_2$$ Given: $m_1 = 5 \text{ kg}$ $u_1 = 10 \text{ m/s}$ $m_2 = 25 \text{ kg}$ $u_2 = 30 \text{ m/s}$ Substitute the values: $$P_{\text{initial}} = (5 \text{ kg})(10 \text{ m/s}) + (25 \text{ kg})(30 \text{ m/s})$$ $$P_{\text{initial}} = 50 \text{ kg} \cdot \text{m/s} + 750 \text{ kg} \cdot \text{m/s}$$ $$P_{\text{initial}} = 800 \text{ kg} \cdot \text{m/s}$$ Step 2: Use the conservation of momentum to find the final velocity. Since the objects coalesce, they move together as a single mass ($m_1 + m_2$) with a common final velocity ($v$). $$P_{\text{final}} = (m_1 + m_2)v$$ By conservation of momentum, $P_{\text{initial}} = P_{\text{final}}$: $$m_1 u_1 + m_2 u_2 = (m_1 + m_2)v$$ $$800 \text{ kg} \cdot \text{m/s} = (5 \text{ kg} + 25 \text{ kg})v$$ $$800 \text{ kg} \cdot \text{m/s} = (30 \text{ kg})v$$ Solve for $v$: $$v = \frac{800 \text{ kg} \cdot \text{m/s}}{30 \text{ kg}}$$ $$v = \frac{80}{3} \text{ m/s} \approx 26.67 \text{ m/s}$$ Step 3: Calculate the total initial kinetic energy. $$KE_{\text{initial}} = \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2$$ $$KE_{\text{initial}} = \frac{1}{2}(5 \text{ kg})(10 \text{ m/s})^2 + \frac{1}{2}(25 \text{ kg})(30 \text{ m/s})^2$$ $$KE_{\text{initial}} = \frac{1}{2}(5 \text{ kg})(100 \text{ m}^2/\text{s}^2) + \frac{1}{2}(25 \text{ kg})(900 \text{ m}^2/\text{s}^2)$$ $$KE_{\text{initial}} = 250 \text{ J} + 11250 \text{ J}$$ $$KE_{\text{initial}} = 11500 \text{ J}$$ Step 4: Calculate the total final kinetic energy. $$KE_{\text{final}} = \frac{1}{2}(m_1 + m_2)v^2$$ $$KE_{\text{final}} = \frac{1}{2}(30 \text{ kg})\left(\frac{80}{3} \text{ m/s}\right)^2$$ $$KE_{\text{final}} = \frac{1}{2}(30 \text{ kg})\left(\frac{6400}{9} \text{ m}^2/\text{s}^2\right)$$ $$KE_{\text{final}} = 15 \text{ kg} \times \frac{6400}{9} \text{ m}^2/\text{s}^2$$ $$KE_{\text{final}} = \frac{96000}{9} \text{ J}$$ $$KE_{\text{final}} = \frac{32000}{3} \text{ J} \approx 10666.67 \text{ J}$$ Step 5: Calculate the loss in kinetic energy. Loss in kinetic energy = $KE_{\text{initial}} - KE_{\text{final}}$ $$\text{Loss} = 11500 \text{ J} - \frac{32000}{3} \text{ J}$$ $$\text{Loss} = \frac{34500}{3} \text{ J} - \frac{32000}{3} \text{ J}$$ $$\text{Loss} = \frac{2500}{3} \text{ J}$$ $$\text{Loss} \approx 833.33 \text{ J}$$ The loss in kinetic energy is $\boxed{\text{}\frac{2500}{3} \text{ J or } 833.33 \text{ J}\text{}}$