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)) ?

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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)) ?

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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 \(p_{train} = m_{train} \times v_{train}\).\(p_{train} = 80 \, \text{kg} \times 4 \, \text{m/s} = 320 \, \text{kg} \cdot \text{m/s}\).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 \(v_f\) be the final velocity of both the train and the cart together.The total mass after the collision is \(m_{train} + m_{cart} = 80 \, \text{kg} + 40 \, \text{kg} = 120 \, \text{kg}\).Step 3: Set up the conservation of momentum equation.\(p_{initial} = p_{final}\)\(320 \, \text{kg} \cdot \text{m/s} = 120 \, \text{kg} \times v_f\)Step 4: Solve for \(v_f\).\(v_f = \frac{320 \, \text{kg} \cdot \text{m/s}}{120 \, \text{kg}}\)\(v_f = \frac{8}{3} \, \text{m/s}\)\(v_f = 2.67 \, \text{m/s}\)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 \(m\), so the mass of the locomotive is \(4m\).Step 1: Since the railroad car is motionless, its initial momentum is 0.Step 2: The locomotive's initial momentum is \(p_{locomotive} = 4m \times v\), where \(v\) is the initial velocity of the locomotive.Step 3: After the collision, the combined mass is \(4m + m = 5m\), and they move together at a velocity \(v_f\).Step 4: Set up the conservation of momentum equation.\(p_{initial} = p_{final}\)\(4m \times v = 5m \times v_f\)Step 5: Solve for \(v_f\).\(v_f = \frac{4m \times v}{5m}\)\(v_f = \frac{4}{5}v\)Answer: The combined speed after the collision is \(\frac{4}{5}v\), where \(v\) 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 \(v\). Then the initial momentum is \(p_{initial} = 2.0 \, \text{kg} \times v\).Step 2: The 1.5 kg mass is motionless, so its initial momentum is 0.Step 3: After the collision, the combined mass is \(2.0 \, \text{kg} + 1.5 \, \text{kg} = 3.5 \, \text{kg}\), and they move together at a velocity \(v_f\).Step 4: Set up the conservation of momentum equation.\(p_{initial} = p_{final}\)\(2.0 \, \text{kg} \times v = 3.5 \, \text{kg} \times v_f\)Step 5: Solve for \(v_f\).\(v_f = \frac{2.0 \, \text{kg} \times v}{3.5 \, \text{kg}}\)\(v_f = \frac{2}{3.5}v\)\( v_f = \

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