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?
Correct: Your answer is correct.
(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?
Correct: Your answer is correct.
(d) What is the x-component of the final velocity of the combined truck-car wreck?
Correct: Your answer is correct.
(e) What impulse did the truck receive from the car during the collision? (Sign matters!)
Incorrect: Your answer is incorrect.
Your response differs from the correct answer by more than 100%.
(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 each of the questions, we need to analyze the conservation of momentum and apply Newton's laws of motion. Here's how you can find the answers step by step:
(a) The total x-component of momentum before the collision can be calculated by adding the individual momentum of each object. Here, the car's momentum is given by mass times velocity (-100.0 kg * -10.0 m/s) and the truck's momentum is given by mass times velocity (4100 kg * 10.0 m/s). Calculate the total momentum by adding these two values.
(b) The x-component of the center-of-mass velocity can be calculated by dividing the total momentum (calculated in part a) by the total mass (900 kg + 4100 kg).
(c) The total x-component of momentum after the collision can be found by using the principle of conservation of momentum. Since it is a perfectly inelastic collision and the two objects move together as one object after the collision, the total momentum before and after the collision remains the same.
(d) The x-component of the final velocity of the combined truck-car wreck can be calculated by dividing the total momentum (calculated in part c) by the total mass (900 kg + 4100 kg).
(e) The impulse experienced by the truck during the collision can be calculated using the equation Impulse = Change in momentum. Since the truck and car move together after the collision, the impulse can be calculated by subtracting the initial momentum of the combined truck-car system (calculated in part a) from the final momentum (calculated in part c).
(f) The impulse experienced by the car during the collision can be calculated in the same way as part e, subtracting the initial momentum of the combined truck-car system from the final momentum.
(g) The average force on the truck from the car during the collision can be calculated using the equation Impulse = Average Force * Time. Rearrange the equation to solve for the average force.
(h) The average force on the car from the truck during the collision can be calculated in the same way as part g.
(i) The impulse experienced by the truck driver from his seatbelt can be calculated using the equation Impulse = Change in momentum. Since the truck and driver move together after the collision, the impulse can be calculated by subtracting the initial momentum of the combined truck-car system (calculated in part a) from the final momentum (calculated in part c).
(j) The impulse experienced by the car driver from his seatbelt can be calculated in the same way as part i.
(k) The average force on the truck driver from the seatbelt can be calculated using the equation Impulse = Average Force * Time. Rearrange the equation to solve for the average force.
(l) The average force on the car driver from the seatbelt can be calculated in the same way as part k.
By following these steps and performing the necessary calculations, you can find the answers to all the questions.