Momentum, Collisions and Explosions Legacy Problem #22 Guided Solution

An effective problem solver reads the problem carefully and develops a mental picture of what's going on, identifies the known and the unknown quantities, and then uses physics principles and conceptual reasoning skills to plot out a strategy as to how to get from the known information to the unknown information. Here we read of a goal line stand and our mental picture is of a player moving one direction towards the goal line and two other players moving the opposite directions in order to stop the original player. We're asked to determine the post-collision speed of the collection of three players after the collision assuming that it's a tackle, that is that Jerome and Michael hang on to the halfback and they travel together at the same speed after the collision. So what we know is that originally the mass of the halfback is 84 kilograms and the halfback is moving south at 6.4 meters per second. Since momentum and velocity are vectors, we need to give attention to this word southward. We also need to give attention to the fact that Jerome and Michael are moving in the opposite direction which we would call northward. Their masses are 102 and 98 kilograms and they're moving at a speed of 3.6 meters per second but the direction is in the opposite direction, northward. The principle we'll use to solve this problem is that the total momentum of both objects before the collision is equal to the total momentum of all objects after the collision. So before the collision, what we have is one object moving south and two objects moving north. So to begin this little solution, we're going to make a definition of what we call positive and negative. And we'll call positive the north direction and negative the south direction. So when we calculate momentum, we'll attach little plus and minus signs to them in order to indicate direction. When we finally get our solution for the final velocity of the objects, it should have a plus minus sign which tells us which direction they're moving. So here, we have 84 kilogram half back moving with a velocity of negative, meaning south, 6.4 meters per second. When you calculate the momentum before the collision of the half back, it's negative 537.6 kilograms per second. You should write that down. Now for Jerome and for Michael, their combined mass is 200 kilograms and they're moving at the same speed, 3.6 meters per second. When you go 200 kilograms times 3.6 meters per second north, you get a positive 720 kilogram meter per second amount of momentum. Now if we find the total system momentum before this collision, we'd have to take the half back's momentum and add to it the linebacker's momentum, keeping in mind that one of them is negative and the other is positive. This total momentum comes out to be 182.40 kilograms meter per second. Now after the collision, the system should still have 182.40 kilograms meter per second of momentum. This momentum is distributed between the three players and all three players are moving with the same velocity. If we were to write their momentum expressions individually, we would call it 84v for the half back, 102v for Jerome, and 98v for Michael. Combined, that's 284 times their velocity, v. So we say 182.40 kilograms meter per second of momentum is equal to 284 kilograms times the velocity, v. Solving for v, we get 0.6423 meters per second comes out to be positive, meaning that the players are moving in the positive or northward direction after the collision.