Calculate the speed of the 1.5 kg trolley at point P using the principle of conservation of mechanical energy.

Step 1: Calculate the speed of the 1.5 kg trolley at point P using the principle of conservation of mechanical energy. Given: Mass of trolley 1, m_1 = 1.5 kg Initial height at A, h_A = 2.0 m Initial velocity at A, v_A = 0 m/s (held stationary) Height at P, h_P = 0 m (bottom of the incline) Acceleration due to gravity, g = 9.8 m/s^2 According to the principle of conservation of mechanical energy: PE_A + KE_A = PE_P + KE_P m_1 g h_A + (1)/(2) m_1 v_A^2 = m_1 g h_P + (1)/(2) m_1 v_P^2 Substitute the known values: (1.5 kg)(9.8 m/s^2)(2.0 m) + (1)/(2) (1.5 kg)(0 m/s)^2 = (1.5 kg)(9.8 m/s^2)(0 m) + (1)/(2) (1.5 kg) v_P^2 29.4 J + 0 = 0 + 0.75 v_P^2 29.4 = 0.75 v_P^2 v_P^2 = (29.4)/(0.75) v_P^2 = 39.2 v_P = sqrt(39.2) v_P ≈ 6.26 m/s The speed of the 1.5 kg trolley at point P is 6.26 m/s. Step 2: Rewrite the complete statement for the principle of conservation of linear momentum. In an isolated system, the total linear momentum is conserved. Step 3: Calculate the speed of the combined trolleys immediately after the collision. This is an inelastic collision where the trolleys stick together. Given: Mass of trolley 1, m_1 = 1.5 kg Velocity of trolley 1 before collision, v_1i = v_P = 6.26 m/s (from Step 1) Mass of trolley 2, m_2 = 2 kg Velocity of trolley 2 before collision, v_2i = 0 m/s (stationary) Combined mass after collision, M = m_1 + m_2 = 1.5 kg + 2 kg = 3.5 kg Let v_f be the final velocity of the combined trolleys. According to the principle of conservation of linear momentum: m_1 v_1i + m_2 v_2i = (m_1 + m_2) v_f Substitute the known values: (1.5 kg)(6.26 m/s) + (2 kg)(0 m/s) = (1.5 kg + 2 kg) v_f 9.39 kg·m/s + 0 = (3.5 kg) v_f 9.39 = 3.5 v_f v_f = (9.39)/(3.5) v_f ≈ 2.68 m/s The speed of the combined trolleys immediately after the collision is 2.68 m/s. Step 4: Calculate the distance travelled by the combined trolleys in 3 s after the collision. After the collision, the combined trolleys move with constant velocity. Given: Velocity of combined trolleys, v_f = 2.68 m/s (from Step 3) Time, t = 3 s Distance travelled, d = v_f × t d = (2.68 m/s)(3 s) d = 8.04 m The distance travelled by the combined trolleys in 3 s after the collision is 8.04 m.