Momentum, Collisions and Explosions Legacy Problem #11 Guided Solution

This is a difficult problem involving a series of three resistive impulses acting upon a car so as to change its momentum from an original value to a final value. What we're asked to determine is the final velocity of the car. Originally, we know the mass of the car, and we know the original velocity. And for each one of these three impulses encountered by the car, we know sufficient amount of information in order to determine the momentum change of the car during that single collision. So, the way we're going to approach this is first determine the original momentum. And then, we're going to begin to subtract momentum from the car. A little bit for each one of the impulses the car encounters. When we're done, we'll have a final momentum. And if we divide that final momentum by the mass, we can figure out the final velocity. So, let's begin by finding the initial momentum of the car before the first impulse. I can do that by taking the car's mass, 1,500 kilograms, and multiplying it by the velocity of 19.8 meters per second. When I do that, I find that the car has an initial momentum of 29,700 units, or kilograms, times a meter per second. The first impulse, we're told, is a 5,700 newton times second impulse, which would cause the same amount of momentum change. So, if I subtract 5,700 kilograms times meter per second from this original momentum, I would have the momentum after the first collision. It would give me a number of 24,000 kilograms times meter per second. The second impulse, we're told, is an impulse that is the result of a 79,000 newton force for 0.12 seconds. I can take the 79,000 newtons and multiply by 0.12, and that would give me an impulse. The value is 9,480 newtons times second. That 9,480 newtons times second impulse would cause a momentum change of 9,480 kilograms times meter per second. So, I'm going to change the momentum again from the 24,000, which we had, to a new value, which is 9,480 units less. When I do that, I get 14,520 units of momentum. We're told that the last impulse causes this 1,500 kilogram car to undergo a velocity change of 3.2 meters per second. If we take that mass and that velocity change, we could calculate a momentum change. 1,500 times 3.2 is 4,800 units of momentum change. Once more, I'm going to subtract this from the value I had after the second impulse. When I do, I get 9,720 kilograms times meter per second as my final momentum. This is the final momentum of the car, and if I divide it by the mass of the car, I'll have the final velocity. So, dividing 9,720 by 1,500 gives me a velocity of 6.48 meters per second. I can round that to 6.5 meters per second, and that would be my answer.