4. Assuming that this is a perfect inelastic collision, calculate the velocity after the collision in the example below. BEFORE AFTER 5. A large locomotive with a mass 4 times that of the smaller motionless railroad car collides and couples together. What is their combined speed after the collision? BEFORE AFTER 6. A 2.0 mathrm(~kg) mass is moving on a frictionless airtrack. It collides into a motionless 1.5 mathrm(~kg) mass. What is the combined speed of the two masses if they stick together on impact? 7. A 1000 mathrm(~kg) car is rolling down the street at 2.5 mathrm(~m) / mathrm(s) . How fast would a 2500 mathrm(~kg) car have to collide into it in order to bring it to rest (0 mathrm(~m) / mathrm(s)) ?

Let's solve each of these problems step by step using the principle of conservation of momentum. The conservation of momentum states that the total momentum of a closed system remains constant if no external forces are acting on it.4. For the first scenario, we have a train with a mass of 80 kg moving at a speed of 4 m/s and a cart with a mass of 40 kg at rest. Since this is a perfectly inelastic collision, the two objects will stick together and move with the same velocity after the collision.Step 1: Calculate the initial momentum of the system.The initial momentum of the train is . .The cart is at rest, so its initial momentum is 0.Step 2: Since momentum is conserved, the total initial momentum is equal to the total final momentum.Let be the final velocity of both the train and the cart together.The total mass after the collision is .Step 3: Set up the conservation of momentum equation. Step 4: Solve for . Answer: The velocity after the collision is 2.67 m/s.5. For the second scenario, we have a large locomotive with a mass 4 times that of the smaller motionless railroad car. Let's denote the mass of the smaller car as , so the mass of the locomotive is .Step 1: Since the railroad car is motionless, its initial momentum is 0.Step 2: The locomotive's initial momentum is , where is the initial velocity of the locomotive.Step 3: After the collision, the combined mass is , and they move together at a velocity .Step 4: Set up the conservation of momentum equation. Step 5: Solve for . Answer: The combined speed after the collision is , where is the initial velocity of the locomotive.6. For the third scenario, we have a 2.0 kg mass moving on a frictionless airtrack and colliding with a motionless 1.5 kg mass.Step 1: Calculate the initial momentum of the system.Let's assume the initial velocity of the 2.0 kg mass is . Then the initial momentum is .Step 2: The 1.5 kg mass is motionless, so its initial momentum is 0.Step 3: After the collision, the combined mass is , and they move together at a velocity .Step 4: Set up the conservation of momentum equation. Step 5: Solve for . \( v_f = \