Proportionality statement manipulations

Adrian_B

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Feb 26, 2021
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Here is the screenshot of questions 15 and 16 from my textbook and the answers from a popular internet resource:
Chapter 5_4 15 and 16.jpg
Answers from an online resource (15c clearly being out of place):
www.chegg.com_homework-help_Applied-Physics-11th-edition-chapter-5.4-problem-15P-solution-9780...jpg
Answer to question 16:
www.chegg.com_homework-help_Applied-Physics-11th-edition-chapter-5.4-problem-16P-solution-9780...jpg
My question is: Why are we allowed to deduce from the final expression that the coefficient of friction is inversely proportional to the stopping distance? Why can we ignore g in the final expression? As well as Vf^2-Vi^2? Can you demonstrate for me in the form of proportionality statements?

If you could also show me where your favorite explanation of this concept can be found I would be appreciative. The best example I am able to find is this youtube video: https: //www.youtube.com/watch?v=zGWMa8bhCLc .Other searches through google only yield proportionality equations and do not show their relationship to proportionality statements. They may say to add a constant when converting to an equation but they do not do what the video does which is what I am looking for (even though the video still falls short for my mind).
 
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