Step 1: Write the principle of conservation of linear momentum. In an isolated system, the total linear momentum remains constant. Step 2: Calculate the velocity of the truck after the collision. Let the car be $c$ and the truck be $t$. Let the initial velocities be $v_i$ and final velocities be $v_f$. Assume the eastern direction is positive. Given: Mass of car, $m_c = 1200$ kg Initial velocity of car, $v_{ci} = +25$ m/s Mass of truck, $m_t = 6000$ kg Initial velocity of truck, $v_{ti} = +15$ m/s Final velocity of car, $v_{cf} = +16$ m/s According to the principle of conservation of linear momentum: $$m_c v_{ci} + m_t v_{ti} = m_c v_{cf} + m_t v_{tf}$$ Substitute the known values: $$(1200 \text{ kg})(25 \text{ m/s}) + (6000 \text{ kg})(15 \text{ m/s}) = (1200 \text{ kg})(16 \text{ m/s}) + (6000 \text{ kg}) v_{tf}$$ $$30000 \text{ kg}\cdot\text{m/s} + 90000 \text{ kg}\cdot\text{m/s} = 19200 \text{ kg}\cdot\text{m/s} + (6000 \text{ kg}) v_{tf}$$ $$120000 \text{ kg}\cdot\text{m/s} = 19200 \text{ kg}\cdot\text{m/s} + (6000 \text{ kg}) v_{tf}$$ $$120000 - 19200 = 6000 v_{tf}$$ $$100800 = 6000 v_{tf}$$ $$v_{tf} = \frac{100800}{6000}$$ $$v_{tf} = 16.8 \text{ m/s}$$ The velocity is positive, indicating it is in the eastern direction. The velocity of the truck after the collision is $\boxed{\text{16.8 m/s to the east}}$. Step 3: Calculate the mechanical energy lost during the car-truck collision. First, calculate the total initial kinetic energy ($KE_i$): $$KE_i = \frac{1}{2} m_c v_{ci}^2 + \frac{1}{2} m_t v_{ti}^2$$ $$KE_i = \frac{1}{2} (1200 \text{ kg}) (25 \text{ m/s})^2 + \frac{1}{2} (6000 \text{ kg}) (15 \text{ m/s})^2$$ $$KE_i = \frac{1}{2} (1200)(625) + \frac{1}{2} (6000)(225)$$ $$KE_i = 375000 \text{ J} + 675000 \text{ J}$$ $$KE_i = 1050000 \text{ J}$$ Next, calculate the total final kinetic energy ($KE_f$): $$KE_f = \frac{1}{2} m_c v_{cf}^2 + \frac{1}{2} m_t v_{tf}^2$$ Using $v_{tf} = 16.8$ m/s from Step 2: $$KE_f = \frac{1}{2} (1200 \text{ kg}) (16 \text{ m/s})^2 + \frac{1}{2} (6000 \text{ kg}) (16.8 \text{ m/s})^2$$ $$KE_f = \frac{1}{2} (1200)(256) + \frac{1}{2} (6000)(282.24)$$ $$KE_f = 153600 \text{ J} + 846720 \text{ J}$$ $$KE_f = 1000320 \text{ J}$$ Finally, calculate the mechanical energy lost: $$\text{Energy lost} = KE_i - KE_f$$ $$\text{Energy lost} = 1050000 \text{ J} - 1000320 \text{ J}$$ $$\text{Energy lost} = 49680 \text{ J}$$ The mechanical energy lost during the collision is $\boxed{\text{49680 J}}$. Step 4: Explain why mechanical energy is lost. Mechanical energy is lost in an inelastic collision because it is converted into other forms of energy, such as heat, sound, and potential energy of deformation (due to the crumpling of the car and truck).