Momentum, Collisions and Explosions Legacy Problem #28 Guided Solution

This is a very difficult problem, a problem that's going to require that you be very organized in your solution, that you be a good thinker, that you draw upon a wealth of physics understanding, and most of all, that you employ the habits of an effective problem solver. I'll talk you through it very extensively. You should write down things as I speak. I call this a momentum plus problem, and in a momentum plus problem, what you have is a collision, and then something that happens either before or after the collision. And what you have to do is solve a momentum problem and combine that solution to a momentum problem with some other form of physical analysis of a pre- or post-collision motion. For instance, here in this problem, what we have is Jack the Ripper throws a pencil at a SpongeBob doll. There's a collision between a pencil and a SpongeBob doll. After the collision, the SpongeBob doll, which was at rest, achieves a post-collision momentum, and thus post-collision velocity. And what we're told is we're told the distance that the SpongeBob doll slides along the countertop before finally stopping. We're given the coefficient of friction, the mu value between countertop and SpongeBob doll, and we're asked to determine the speed at which the pencil was moving before striking SpongeBob. So the two parts of the problem is obviously the collision part, in which you relate pre-collision momentum to post-collision momentum. And then the other part of the problem is the sliding of the SpongeBob doll across the countertop. We have to approach the second part of the motion first and sort of work backwards towards determining the pre-collision speed of the pencil. So I'm going to begin by dividing my sheet of paper, my solution, into two parts with a vertical line separating the two parts. The second part being the second part of the motion is the sliding of the SpongeBob doll to a stop from some original speed occurring after the collision to a final speed of zero over a distance of 11.9 centimeters. That's a nasty unit on distance, so right away I'm going to convert that to a distance of 0.119 meters. Now if I can find the collision or the speed of the SpongeBob doll and pencil embedded within it just after the collision, before this deceleration period, then I can use that in the other half of my solution, which would be the half in which I determine the pre-collision velocity of the pencil. So that's going to be my overall strategy. So I'm going to analyze this post-collision motion. I know two things right now, that the SpongeBob doll and pencil has a final velocity of zero and it moved a d of 0.119 meters. Now I'm thinking kinematic equations and Newton's laws, and I'm thinking I can get the original velocity of the SpongeBob doll before sliding to stop if I can just know one more quantity. I've not yet used the coefficient of friction, 0.325, so I'm presuming I need to use that. I can use that to find the acceleration. It goes something like this. I draw a free body diagram for the SpongeBob doll and pencil. I draw a force down. It's gravity. It's value is m times g. I draw a force up that's the normal force of the countertop pushing up on SpongeBob. It's value is going to be the same as the down force, so it would be mg as well. Then I draw a backwards force on the SpongeBob doll, and that's friction. That backwards force of friction is what slows it down. It's the net force. So f net is the friction force, and f friction is simply mu times f norm, or mu times mg. Now that mu times mg, the net force or friction force, is equal to ma. So I say mu mg equals ma, and then I cancel my m's, and I'm left with the a is equal to mu times g. Now I can substitute in 0.325 for the value of mu, and 9.8 for the value of g, and I can get an acceleration. It comes out to be 3.1850 meters per second squared. Now I can take this value for acceleration and use it along with a vf of 0 and a d of 0.119 meters in order to solve for the velocity of the SpongeBob doll and pencil before finally skidding to a stop. I would use the equation vf squared equals vo squared plus 2ad. When I substitute 0 in for vf squared, and I substitute 3.1850 meters per second squared negative in for a, and I substitute 0.119 meters in for d, I end up with a velocity of 0.75803 meters per second. And that was the velocity before it began skidding to a stop. We can also call that the velocity of the SpongeBob doll and pencil immediately after the collision. So now I'm ready to do the momentum analysis for this problem. And in the momentum analysis, what I do is I set the pre-collision momentum of SpongeBob doll and pencil equal to the post-collision momentum of SpongeBob doll and pencil. I would say 4.0 grams times v plus 221 times 0 is equal to the 225 grams, that's the combined mass of the pencil and SpongeBob, times 0.75803. That's the post-collision velocity that I've determined by doing my kinematic analysis. Now I'm going to simplify that to 4v is equal to, and the right side evaluates to, 170.55675. I'll divide through by 4, and I'll have v equal 42.64 meters per second, or 43 meters per second when rounded to two digits. That's equivalent to 96 miles per hour, and that explains why they call this boy Jack the Ripper.