Momentum, Collisions and Explosions Legacy Problem #26 Guided Solution

A good problem solver reads the problem carefully and begins to develop a mental picture of what's going on. You'll notice that I have developed my mental picture using a diagram of the two objects involved in the collision. I've shown the pre- and post-collision information about the object. A good problem solver also identifies known information and writes it down. A diagram is a great place to include this known information. A good problem solver identifies the unknown quantity and then begins to use physics principles and conceptual reasoning skills to plot out a strategy as to how to get from the known to the unknown information. Here I'm going to do that. I'm going to begin by writing down what I know. And so on my diagram, you'll notice that I've recorded the mass of cart A and the mass of cart B. I'm showing it before the collision and also after the collision. I've shown that cart A is originally moving 27.6 centimeters per second to the right. And I've shown that cart B originally is moving at 42.9 centimeters per second to the left. I'm going to have to keep track of momentum in this problem, and since it's a vector and based upon velocity, I'm going to have to include plus and minus signs for my velocity and my momentum quantities. Thus, I'm going to define to the right as positive and to the left as negative. And so I've recorded the velocity of cart B before the collision as being negative 42.9 centimeters per second. After the collision, we're told that cart A turns around and moves back to the left at 10.1 centimeters per second. The fact that that's a leftward velocity and the fact that I've defined left as negative means that that post-collision velocity for cart A is negative 10.1 centimeters per second. And of course, the mass of cart A has not changed. It's still 1.0 kilograms. We want to know what's the velocity of cart B after the collision. I don't know whether cart B is moving to the left or to the right. I'm going to let the mathematics determine the answer to that question. If the value of v comes out to be positive, then I know cart B is moving to the right after the collision. And if it comes out to be negative, I know that cart B is moving to the left after the collision. The principle I use to determine this post-collision velocity will be the principle that the pre-collision momentum is equal to the post-collision momentum. Not the pre-collision momentum of cart A or cart B, but of the combination of the two carts, A and B. We sometimes refer to that as a system. So I'm going to write a statement. You ought to follow along and write it down as I speak it out. Let the momentum of A plus the momentum of B before the collision equal the momentum of A plus the momentum of B after the collision. Numerically it comes out to be like this. 1.0 times 27.6, that's from cart A before, plus 0.5 times negative 42.9, that's the momentum of cart B before, is equal to, now comes the after-collision moment, is equal to 1.0 kilograms times negative 10.1. That's cart A afterwards. Plus 0.50 times V, where the V is my post-collision velocity of cart B, the unknown I'm trying to solve for. Hey, there you have it. One equation with one unknown in it. Theoretically we should be able to solve for that unknown. Doing so means we have to use good algebra skills. I'm going to begin by simplifying the left side of the equation. It comes out to be 27.6 for cart A plus negative 21.45 for cart B. Summing these two numbers gives me 6.15 kilograms times a centimeter per second. That's the left side of the equation. The right side is simply negative 10.1 plus 0.5B. Now if I get the 0.5B by itself, I would have to add 10.1 to each side of the equation. Doing so gives me 16.25 kilograms centimeter per second is equal to 0.50 times V. Dividing through by 0.50 kilograms gives me the answer to this question, 32.5 centimeters per second. It comes out to be positive, indicating to me that the direction that cart B is moving after the collision is to the right.