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- [Voiceover] Alright, now let's tackle part c. Use quantitative reasoning, including equations as needed, to develop an expression for the new final position of the block. Express your answer in terms of D. Alright, I'm gonna set up a little table here for part c. Oops, sorry about that. Sometimes my pen is not functioning properly. Alright so part c. Let me set up two scenarios. So I have scenario one. Scenario one, where we compress the spring, our delta x is equal to d, and then we have scenario two, scenario two, where we compress the spring by twice as much, is equal to two d, and let me set up my table now, so, just like that and like this, and let's just think about a few things. So the first thing I wanna think about is the potential energy, so the potential energy when the spring is compressed. So potential energy when spring compressed. When, I'll just write compressed, when compressed. Well in this scenario I'll call it the potential energy for scenario one, it's equal to one half times the spring constant, times how much we compress it squared. Now what about scenario two? This potential energy is going to be equal to one half times the spring constant, times how much we compress it, it's now twice as much, squared, well this is equal to one half times the spring constant, times four d squared, and I can out the four up front, this is equal to four times one half, four times one half, times our spring constant, times d squared, which is equal to four times the potential energy when we just compress the spring by d. So we already see a little bit of what we talked about in part b, you compress your spring twice as much, you're going to have four times the potential energy, because the potential energy doesn't grow proportionately with how much you compress it, it grows with the square of how much you compress it. Alright, now let's think about kinetic energy Kinetic energy when x is equal to zero, so right when the spring, when we lose contact with the spring, the spring is no longer pushing on the block, well our kinetic energy is going to be equal to what our potential energy was when the spring was actually compressed. Another way of thinking about it, all that potential energy has now been turned into kinetic energy. Now what about over here? Well, the kinetic energy in this scenario, like we just saw before, that's gonna be equal to the potential energy when the spring was compressed. All of that potential energy gets turned into kinetic energy and this is equal to four times u one, four times the potential energy in scenario one, which is the same thing as four times, which is equal to four times the kinetic energy in scenario one. So we have four times the kinetic energy. Four times kinetic energy, kinetic, kinetic energy. So then we have stopping distance, stopping, stopping distance. We know here this is three d and we know, and then we can say "What's this?", this question mark. Well let's just think a little bit about this. We know that if we have that kinetic energy at x equals zero, so we know that k one plus the work done by friction, so let me make it clear, this right over here, that is work done by friction, work done by friction, and it's gonna be negative work cause the force of friction is acting in the direction opposite of the change in x, so the kinetic energy plus the work done by friction is going to be equal to zero. This work cancels out all of this energy. When we think about it, it's turning it all into heat. And so let's think about what the work done by friction is equal to. Well the work done by friction is equal to, is equal to, the coefficient of friction times the mass of the block, times the gravitational field, times how far, over what distance that force, this right over here is the force of friction, times over what distance that force was applied, so times three d. And to be clear, this force is going in the opposite direction of our change in x, so because of that this will be a negative, and so we can say, we can say that the kinetic energy at x equals zero, and now I can just write it as minus mu, the coefficient of friction, times mass, times the gravitational field, times three d is equal to zero. We can add this to both sides and we can get k one is equal to mu times m times g times three d, and if you wanted to solve for distance here, you can divide both sides by the force of friction. So divide both sides by mu times m times g, and you get three d, and I'm just swapping the sides here, is going to be equal to the amount of kinetic energy we have right as x equals zero, divided by mu times m times g and you can just view this as the force of friction. The force, I'll just call it the force of friction right over there. So if you wanna figure out your stopping distance you just figure out your kinetic energy, right when you, right at x equals zero, right when you start entering into the frictiony part of your platform and then you divide that by the force of friction, and that will give you your distance traveled. So the distance here, distance, so I can just put some arrows right over here, our distance is going to be equal to k two divided by the force of, the force of friction. Well k two is equal to four times k one, is equal to four times k one. and our force of friction is going to be the same We have the same coefficient of friction, we have the same mass, we have the same gravitational field, so divided by force of friction, and this we already know, k one divided by force of friction is equal to three d, so this is all going to be equal to four, this is going to be four times three d, three d, which is equal to 12 d, 12 d. So this is all a mathematical way of saying you compress it twice as much, you're going to have four times the potential energy when your spring is compressed, which means you're going to have four times the kinetic energy at x equals zero, which means it is going to take, you're going to have four times the stopping distance, so instead of stopping at three d or in six d as what the student proposed, you are now stopping at 12 d, so that is our stopping distance. Did we answer all of, yeah we answered all of part c. Use quantitative reasoning, including equations as needed, to develop an expression for the new final position of the block. Express your answer in terms of d. Yep, feel good about that.