Momentum, Collisions and Explosions Legacy Problem #18 Guided Solution

A good problem solver has a habit of reading the problem carefully and getting a mental picture of what's going on, identifying known and unknown quantities, and using conceptual reasoning skills to get from the known information to the unknown information. Here the picture we have is of a ball moving one direction and hitting a racket moving the opposite direction. And what we're told is the mass of the ball and the mass of the racket. We're told the speed at which the ball is moving forward before the collision and the speed at which it's moving in the opposite direction after the collision. This is sufficient enough information for us to find the velocity change of the ball and the momentum change of the ball. We're guided through the process here in a part A, B, C, D type situation. So the first question they ask is to determine the pre-collision momentum of the ball. The pre-collision momentum of the ball would be the speed of the mass of the ball multiplied by the velocity of the ball. So we'll take 57.5 grams as the mass of the ball, and 26.7 meters per second north is the velocity of the ball, and we'll multiply and we get the momentum of the ball before the collision. It comes out to be 1535.25 grams times meter per second north. Now the second part of the problem is to find the post-collision momentum of the ball, that is after the collision with the racket. We're given the speed of the ball after that collision, it's 29.5 meters per second, and we know that the mass of the ball was 57.5 grams. So multiplying mass by velocity gives us the momentum, and when you do that it comes out to be 1696.25 grams times meter per second south. Now you'll notice that before the collision with the racket the ball was moving north, and afterwards it was moving south. It had northward momentum, and now it has southward momentum. When we get around to do part C, where we wish to find the momentum change, we have to consider direction, because the fact that it was moving north, slowed down to zero, and then moved back south, means that its momentum change went from a positive to a negative value. And to find that change, you have to call either the north or the south negative, and then do the final minus the initial value. So what I'm going to do is I'm just going to say that the south is negative. And so when I go to calculate the momentum change, I would go negative 1696.25 minus 1535.25. Say that again, I'm going to call south negative, and then I'm going to say the negative 1696.25 is equal to, or minus the 1535.25 is equal to the momentum change. Turns out to be a negative 3231.5 grams meter per second. Now the fact that it's negative means that it's south, because that's how we've defined negative is as south. So the momentum change is 3231.5 grams meter per second. That's momentum change of the ball, and it's also momentum change of the racket. So if we wish to answer question D, which is find the velocity change of the racket, we merely need to employ the principle that the momentum change of one object is equal and opposite to the momentum change of the other object. So the momentum change of the racket is 3231.5 grams meter per second north, and if we divide 331 grams into this figure, we'll get the velocity change of the racket comes out to be 9.7628 meters per second north. And we can just round that to 9.8 meters per second north.