Most of us know intuitively that in a head-on collision between a large dump truck and a subcompact car, you are better off being in the truck than in the car. Why is this? Many people imagine that the collision force exerted on the car is much greater than that experienced by the truck. To substantiate this view, they point out that the car is crushed, whereas the truck in only dented. This idea of unequal forces, of course, is false. Newton's third law tells us that both objects experience forces of the same magnitude. The truck suffers less damage because it is made of stronger metal. But what about the two drivers? Do they experience the same forces? To answer this question, suppose that each vehicle is initially moving at 10.0 m/s and that they undergo a perfectly inelastic head-on collision. (In an inelastic collision, the two objects move together as one object after the collision.) Each driver has a mass of 100.0 kg. Including the drivers, the total vehicle masses are 900 kg for the car and 4100 kg for the truck. The collision time is 0.090 s. Choose coordinates such that the truck is initially moving in the positive x direction, and the car is initially moving in the negative x direction.
(a) What is the total x-component of momentum BEFORE the collision?
(b) What is the x-component of the CENTER-OF-MASS velocity BEFORE the collision?
(c) What is the total x-component of momentum AFTER the collision?
(d) What is the x-component of the final velocity of the combined truck-car wreck?
(e) What impulse did the truck receive from the car during the collision? (Sign matters!)
(f) What impulse did the car receive from the truck during the collision? (Sign matters!)
(g) What is the average force on the truck from the car during the collision? (Sign matters!)
(h) What is the average force on the car from the truck during the collision? (Sign matters!)
(i) What impulse did the truck driver experience from his seatbelt? (Sign matters!)
(j) What impulse did the car driver experience from his seatbelt? (Sign matters!)
(k) What is the average force on the truck driver from the seatbelt? (Sign matters!)
(l) What is the average force on the car driver from the seatbelt? (Sign matters!)
To answer these questions, we need to use the principles of conservation of momentum and Newton's laws of motion. Let's go through each part:
(a) The total x-component of momentum before the collision can be calculated by multiplying the mass of each object by its initial velocity and summing them up. The total momentum is given by:
Momentum_car = mass_car * velocity_car
Momentum_truck = mass_truck * velocity_truck
Total_momentum_before = Momentum_car + Momentum_truck
(b) The x-component of the center-of-mass velocity before the collision can be calculated by dividing the total momentum before the collision by the total mass of the system. The center-of-mass velocity is given by:
Center_of_mass_velocity_x_before = Total_momentum_before / (mass_car + mass_truck)
(c) The total x-component of momentum after the collision can be calculated by considering that the car and truck move together as one object. Therefore, their combined mass will be the sum of the individual masses. The total momentum is given by:
Total_momentum_after = (mass_car + mass_truck) * final_velocity_combined
(d) The x-component of the final velocity of the combined truck-car wreck can be calculated by dividing the total momentum after the collision by the total mass of the system:
final_velocity_combined = Total_momentum_after / (mass_car + mass_truck)
(e) The impulse experienced by the truck during the collision can be calculated by subtracting the initial momentum from the final momentum of the truck:
Impulse_truck = mass_truck * (final_velocity_combined - velocity_truck)
(f) The impulse experienced by the car during the collision can be calculated by subtracting the initial momentum from the final momentum of the car:
Impulse_car = mass_car * (final_velocity_combined - velocity_car)
(g) The average force on the truck from the car during the collision can be calculated by dividing the impulse experienced by the truck by the collision time:
Average_force_truck = Impulse_truck / collision_time
(h) The average force on the car from the truck during the collision can be calculated by dividing the impulse experienced by the car by the collision time:
Average_force_car = Impulse_car / collision_time
(i) The impulse experienced by the truck driver from his seatbelt can be calculated by considering the change in momentum of the truck driver. Since we have the mass and final velocity of the truck, we can calculate the initial velocity of the driver using the principle of conservation of momentum. Then the impulse can be calculated as:
Impulse_truck_driver = mass_truck_driver * (final_velocity_combined - initial_velocity_truck_driver)
(j) The impulse experienced by the car driver from his seatbelt can be calculated in a similar way as the impulse experienced by the truck driver. We need to calculate the initial velocity of the car driver using the principle of conservation of momentum and then calculate the impulse as:
Impulse_car_driver = mass_car_driver * (final_velocity_combined - initial_velocity_car_driver)
(k) The average force on the truck driver from the seatbelt can be calculated by dividing the impulse experienced by the truck driver by the collision time:
Average_force_truck_driver = Impulse_truck_driver / collision_time
(l) The average force on the car driver from the seatbelt can be calculated by dividing the impulse experienced by the car driver by the collision time:
Average_force_car_driver = Impulse_car_driver / collision_time
By solving these calculations, we can obtain the values for each of the given quantities.