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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 going to set up a little table here for Part C whoops sorry about that sometimes my pin 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 to scenario two where we compress the spring by twice as much is equal to 2 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 want to think about is the potential energy so the potential energy when the spring is compressed so potential potential energy when spring compressed when I'll just write compressed when compressed well in this area scenario I'll call that potential energy for scenario 1 it's equal to 1/2 times the spring constant times how much we compress it squared now what about scenario 2 this potential energy is going to be equal to 1/2 times the spring constant times how much we compress it it's now twice as much squared well this is equal to 1/2 times the spring constant times 4 d squared and I could put the 4 out front this is equal to 4 times 1/2 4 times 1/2 times our spring constant times d squared which is equal to 4 times the potential energy when we just compressed the spring by D so we already see a little bit of what we talked about in Part B you compress just 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 you you compress it and grows with the square of how much you compress it all right now let's think about let's think about kinetic energy kinetic 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 while our kinetic energy is going to be equal to what our potential energy was when the spring was actually compressed all over 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 going to be equal to the potential energy when the spring was compressed where all of that potential energy gets turned into kinetic energy and this is equal to four times u1 four times the potential energy and 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 3d and that we know heat and then we can say well what is 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 k1 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 going to be negative work because 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 it's one way to think about its 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 they that over what distance that force this right over here is a 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 of our change in X so because of that this will be a 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 get K 1 is equal to MU times M times G times 3 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 3 D and I'm just swapping the sides here is going to be equal to the amount of kinetic energy we have right at x equals zero divided by mu times M times G and you could 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 want to figure out your stopping distance you just figure out your kinetic energy right when you write it x equals zero right when you start entering into the friction each part of your 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 k2 divided by the force of the force of friction well k2 is equal to four times k1 is equal to four times k1 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 this we already know K 1 divided by force of friction is equal to 3 D so this is all going to be equal to 4 this is going to be 4 times 3 D 3 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 4 times the potential energy when your spring is compressed which means you're going to have 4 times the kinetic energy at x equals 0 which means it's going to take you're going to have 4 times the stopping distance so instead of stopping at 3 D or + 6 D is 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