Here are the calculations for part (c): Given: Mass of object 1, $m_1 = 8 \, \text{kg}$ Initial velocity of object 1, $u_1 = 50 \, \text{m/s}$ Mass of object 2, $m_2 = 2 \, \text{kg}$ Initial velocity of object 2, $u_2 = -30 \, \text{m/s}$ (negative sign indicates opposite direction) After impact, the objects move together with a common velocity $V$. i. The value of V: Step 1: Apply the principle of conservation of momentum. For an inelastic collision where objects stick together, the total momentum before the collision equals the total momentum after the collision. $$m_1 u_1 + m_2 u_2 = (m_1 + m_2)V$$ Step 2: Substitute the given values into the equation. $$(8 \, \text{kg})(50 \, \text{m/s}) + (2 \, \text{kg})(-30 \, \text{m/s}) = (8 \, \text{kg} + 2 \, \text{kg})V$$ $$400 \, \text{kg} \cdot \text{m/s} - 60 \, \text{kg} \cdot \text{m/s} = (10 \, \text{kg})V$$ $$340 \, \text{kg} \cdot \text{m/s} = (10 \, \text{kg})V$$ Step 3: Solve for $V$. $$V = \frac{340 \, \text{kg} \cdot \text{m/s}}{10 \, \text{kg}}$$ $$V = 34 \, \text{m/s}$$ The common velocity after impact is $\boxed{\text{34 m/s}}$. ii. The loss in kinetic energy in the process: Step 1: Calculate the total initial kinetic energy ($KE_{initial}$). $$KE_{initial} = \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2$$ $$KE_{initial} = \frac{1}{2}(8 \, \text{kg})(50 \, \text{m/s})^2 + \frac{1}{2}(2 \, \text{kg})(-30 \, \text{m/s})^2$$ $$KE_{initial} = \frac{1}{2}(8)(2500) + \frac{1}{2}(2)(900)$$ $$KE_{initial} = 4(2500) + 1(900)$$ $$KE_{initial} = 10000 \, \text{J} + 900 \, \text{J}$$ $$KE_{initial} = 10900 \, \text{J}$$ Step 2: Calculate the total final kinetic energy ($KE_{final}$). $$KE_{final} = \frac{1}{2}(m_1 + m_2)V^2$$ $$KE_{final} = \frac{1}{2}(8 \, \text{kg} + 2 \, \text{kg})(34 \, \text{m/s})^2$$ $$KE_{final} = \frac{1}{2}(10 \, \text{kg})(1156 \, \text{m}^2/\text{s}^2)$$ $$KE_{final} = 5(1156)$$ $$KE_{final} = 5780 \, \text{J}$$ Step 3: Calculate the loss in kinetic energy. $$\text{Loss in KE} = KE_{initial} - KE_{final}$$ $$\text{Loss in KE} = 10900 \, \text{J} - 5780 \, \text{J}$$ $$\text{Loss in KE} = 5120 \, \text{J}$$ The loss in kinetic energy is $\boxed{\text{5120 J}}$.