Newton's Laws Legacy Problem #30 Guided Solution

This is a very difficult problem, and it's going to demand that you practice the habits of an effective problem solver, that you draw on your conceptual understanding of the relationship between quantities, and that you use your noodle, not your brain. So, I'm going to begin by reading the problem. We have a car, mass is known, 1370 kilograms. I'm going to write m equal 1370 kilograms. It's skidding to a stop, and that's going to tell me something right there. It's going to tell me that the wheels are no longer turning, and it's skidding to a stop. We're told that the car decelerates from 27.6 meters per second to a rest position, and that tells me information about an original and a final speed. And this change in speed occurs over the course of 3.15 seconds. So, I see three pieces of kinematic information there, and I'm going to write them down. V0 equals 27.6 meters per second, Vf equals 0 meters per second, and T equals 3.15 seconds. And it says, assume negligible air resistance and determine the coefficient of friction between the cars and the road surface. So, what it's asking me to calculate is mu. Find out what mu is. Okay, so now as I look at what I know and what I'm trying to find, my mind kind of races around, and I think, how can I do this? I know I'm probably going to have to use a free-body diagram, and I know that I'm going to have to use the f frict equal mu f norm equation. So, it's the only equation I can think of that has this coefficient of friction, this mu thing in it. Now, if I think about that a little bit more, I would reason that to find mu, I'm going to need to know two things, a friction force and the f normal force. And it's going to demand that I draw a free-body diagram for the skidding car. So, I'm going to do that real quick. There's a gravity force down, a normal force up, and a friction force to the left, presuming a rightward motion. Now, the ups and downs, they've got to balance. That's how I'm thinking through it. The ups and downs, they have to balance, but the friction force, there's nothing to balance it. So, that's got to be the net force. I'm looking for two things, f frict and f norm. So, if that's the case, f frict, that's got to be the net force. That's got to be ma. So, how do I calculate ma? Well, the m part is easy. It's given as 1370 kilograms. But the a part, that's going to take the use of a kinematic equation in the three kinematic quantities. So, let's talk that one through. How would I get a from vf, vo, and t? Well, that's pretty easy. You use the equation vf equals vo plus at, where you simply use the relationship that acceleration is the delta v per t. The delta v here is the change from 27.6 to 0. That's a negative 27.6 meters per second change. Divide that by the time, and you get the acceleration. Why did we need that? Because we needed to get f net, because f net was that friction. So, now that you have acceleration, calculate the f net. You get a negative a and a negative f net, and what that simply means is that the force is against the motion of the object. The friction force goes left as the object skids to the right, and so now I know f friction. Why did I need to know that? Well, because I want to calculate mu as the ratio of f fric to f norm. I need that numerator, and I also need the denominator. So, look at your free body diagram, which I hope you've drawn. There's two vertical forces, f grab down, which can be calculated as g, and f norm up, which is equal to f grab down, since no vertical acceleration means those forces must be equal and opposite. So, now I take that f grab, I say it's equal to f norm, and I plug f norm value into the denominator, and I calculate mu as the ratio of f fric to f norm. Wow, that's a lot of thinking, and a pretty good illustration of how you would use the habits of an effective problem solver to take the given information, the description of the scenario, and use it to work your way through to a solution of a problem.