Write the principle of conservation of linear momentum.

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 kg)(25 m/s) + (6000 kg)(15 m/s) = (1200 kg)(16 m/s) + (6000 kg) v_tf 30000 kg·m/s + 90000 kg·m/s = 19200 kg·m/s + (6000 kg) v_tf 120000 kg·m/s = 19200 kg·m/s + (6000 kg) v_tf 120000 - 19200 = 6000 v_tf 100800 = 6000 v_tf v_tf = (100800)/(6000) v_tf = 16.8 m/s The velocity is positive, indicating it is in the eastern direction. The velocity of the truck after the collision is 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 = (1)/(2) m_c v_ci^2 + (1)/(2) m_t v_ti^2 KE_i = (1)/(2) (1200 kg) (25 m/s)^2 + (1)/(2) (6000 kg) (15 m/s)^2 KE_i = (1)/(2) (1200)(625) + (1)/(2) (6000)(225) KE_i = 375000 J + 675000 J KE_i = 1050000 J Next, calculate the total final kinetic energy (KE_f): KE_f = (1)/(2) m_c v_cf^2 + (1)/(2) m_t v_tf^2 Using v_tf = 16.8 m/s from Step 2: KE_f = (1)/(2) (1200 kg) (16 m/s)^2 + (1)/(2) (6000 kg) (16.8 m/s)^2 KE_f = (1)/(2) (1200)(256) + (1)/(2) (6000)(282.24) KE_f = 153600 J + 846720 J KE_f = 1000320 J Finally, calculate the mechanical energy lost: Energy lost = KE_i - KE_f Energy lost = 1050000 J - 1000320 J Energy lost = 49680 J The mechanical energy lost during the collision is 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).