Momentum, Collisions and Explosions Legacy Problem #27 Guided Solution

A good problem solver reads the problem carefully and develops a mental picture of what's going on, identifying 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. As we read this problem, we read that there's a collision between a ball, a bowling ball, and a bowling pin. The best way to diagram the situation is to simply draw a pre-collision representation of the ball and the pin, and then a post-collision representation of the ball and the pin. In the diagram, record the masses of the ball and the pin, and record the velocities that are known before and after the collision. If I were to do that, I would have a diagram in which, before the collision, I show a ball moving to the right at 8.24 meters per second. Its mass is 7.05 kilograms. And then I would draw a pin, and its mass is 1.52 kilograms. And though not stated, we know from the context that its velocity is 0 meters per second. Now, after the collision, the ball continues moving forward, and the unknown quantity I wish to calculate is the velocity of this 7.05 kilogram ball. The pin is said to be flying forward at 13.2 meters per second. That's its velocity. I'd record that on the diagram. And its mass remains as 1.52 kilograms. The principle I use to get from this known information to the unknown quantity of the post-collision sphere of the ball is the principle that the total momentum of the system before the collision is equal to the total momentum of the system after the collision. That is, that the momentum of the ball plus the momentum of the pin before the collision is equal to the momentum of the ball plus the momentum of the pin afterwards. So I'm going to write a mathematical statement, and you should follow along and write it as we go. The statement goes like this. 7.05 kilograms times 8.24 meters per second. That's the momentum of the ball before the collision. Plus 1.52 kilograms times 0 meters per second. That's the momentum of the pin before the collision. It doesn't have any. Is equal to 7.05 kilograms times v. That's the momentum of the ball after the collision. We don't know the velocity, so we're just calling it v. Plus 1.52 kilograms times 13.2 meters per second. So what I've written is 7.05 times 8.24 plus 1.52 times 0 equal 7.05 times v plus 1.52 times 13.2. Hey, that's one equation on one unknown, and I should be able to solve for the unknown. Solving for the unknown demands that you now use good algebra skills. You've transformed the verbal statement of the problem into a mathematical exercise. The left side of the equation simplifies to 58.092 kilograms times meters per second. And the right side becomes 7.05v plus 20.064 kilograms meters per second. Subtracting the 20.064 kilograms meters per second from each side gets my 7.05v prime by itself. I would have 38.028 equals 7.05v prime. And dividing through by the 7.05 kilograms gives me as my final velocity value for the wall. That's 5.394 meters per second. I'd round that to three significant digits.