Always show substitutions whether whole equations

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*Always show substitutions, whether whole equations, constants, zeros, or any numbers. Example: Solve for ? as a function of ? and t with ? 0 = 0 given: ? = ?? and ? = ? 0 ? + 1 2 𝑎? 2 and ? = ? 0 + 𝑎? Choose starting equation that contains desired variable: ? = ?? Need to find a way to get x into the equation. The second equation listed has x in it, but does not have v, so substitute in the third equation to eliminate v: Since ? = ? 0 + 𝑎? Then ? = ?(? 0 + 𝑎?) As defined in the problem, ? 0 = 0 , so substitute in to simplify: ? = ?(0 + 𝑎?) = ?𝑎? Still need to get x involved, so use second equation: Let ? = ? 0 ? + 1 2 𝑎? 2 As defined in the problem, ? 0 = 0 , so substitute in to simplify: ? = (0)? + 1 2 𝑎? 2 = 1 2 𝑎? 2 In many cases, t is an undefined variable because it is difficult to measure with precision. While not required for working the example problems, I will choose to eliminate t from the equation. You will learn how to identify from a problem which variables are or are not allowed in a final answer. Solve the x equation for t: Since ? = 1 2 𝑎? 2 Then ? = √ 2𝑥 𝑎 Substitute in to the equation for p: Therefore ? = ?𝑎? = ?𝑎√ 2𝑥 𝑎 At this point we could rewrite and box this solution, since it meets the requirements of the problem, or we could simplify by combining the a terms. Simplification is not always necessary unless you are trying to match your algebra to the solutions in a multiple choice question. However, to practice some simplification skills, we will simplify this result below. Recall that 𝑎 = 𝑎 1 = √ 𝑎 2 1 2 = √ 𝑎 2 1 therefore we can simplify by: ? = ?𝑎√ 2𝑥 𝑎 = ?√ 𝑎 2 1 2𝑥 𝑎 Recall that √? √? = √?? therefore we can further simplify by: ? = ?√ 𝑎 2 1 2𝑥 𝑎 = ?√2𝑎? Always rewrite and box the final answer: ? = ?√2𝑎?
Note: If I was working this problem on an AP exam, there are a few shortcuts for notation of substitutions that I would have used that will be demonstrated for you in class, and my presentation of steps would not have been quite so formal or wordy, and I would have stopped at the unsimplified solution to save time. For those of you who are naturally quick with math and can do the algebra quickly in your head, it can more time to write out the necessary work than it took you to solve the problem, and for those of you who show your work, but chaotically, you may struggle to keep your work organized enough to show your train of thought coherently on paper. However, your work is your evidence of your thought process. The free response section of the AP exam, where you are required to show your work, is designed to test your understanding of physics and your ability to communicate the process of arriving at the correct solution by the correct application of math and physics concepts, not to see if you can get the right answer. You need to practice showing your work appropriately now as you will be graded using AP-style rubrics on all of your exams and homework assignments.

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