05 the development of a kinematics problem. February 07, Area under the curve

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1 Area under the curve Area under the curve refers from the region the line (curve) to the x axis 1

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4 From Graphs to equations Case 1 scatter plot reveals no apparent relationship Types of equations Case 2 d t plot suggest a linear relationship Equations Case 3 d t plot suggests an exponential pattern Types of equations Equations Keys to problem solving 1. Draw the details of the question including the question details. 2. Determine the problem type 3. Determine the fundamental equations 4. List all the variables and indicate those that are indicated in the questions. look for hints such as v 1 =0, v d = 0 or Δd = 0 5. If you have 1 equation, you can solve for 1 variable. If you have 2 equations, you can find unique solutions for variables. If you have 3 equations, you can find unique solutions for variables 6. Check for SI units 7. Check for directional arguments (in 2 dimensions, + and, can be used to indicate direction. 8. Plug and solve. Make sure your answer contains the appropriate units and a direction is indicated if a vector. Problems page 8, page 18 #1 7 (3 is tough), page 23 4

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7 The story so far... Day 1 At first, science ( to know) was seen as a study of truths in the world. Early scientists looked at natural patterns to discover "truths". Galileo argued that we need to quantify these truths. Bacon argued that we needed to make multiple observations to verify the pattern. Descartes argued that we need to use deductive reasoning to verify these patterns. Today, a combination of all early works were combined to establish the "scientific method" Note patterns that were extremely consistent lead to the establishment of laws things that were always true ex. Today however, the term law is rarely, if ever, tagged to a pattern. Scientists realize the world is far more complex than first believed. Day 2/3 from the search for patterns through experiment, the need to measure was established. Over time, the metric system was developed to organize and standardize these measures. SI units and significant digits reflect the structure involved in measurement. This structure creates a range of acceptable answers regarding the validity of a claim when performing an experiment. These rules are difficult for high school students to fully understand so the high school student should be careful about how many digits they report in an answer as the number of decision reflects the certainty of the answer. Ideas precision, accuracy, significant digits, procedural and instrumental uncertainty, error calculation Day 3/4 in the search for a relationship, specifically a mathematical relationship, a graph a 2 dimensional graph played a critical role in the analysis of this relationship. (experiments are designed to compare 2 variables at a time Newton and his 2nd law) From a general graph information can be gathered in 1 of 3 ways 1. reading the graph interpolation, extrapolation (both of which can be done through an equation) 2. Slope 3. Area under the curve Slope and area become more difficult if the relationship is not linear. When not linear restrictions are placed on the information obtained from the graph ( average and instantaneous) When linear slope is constant (any two points (or 1 if (0,0) is a point)), Area is a determined through geometric shapes (but the units dictate the units of the outcome) When not linear different techniques are required Slope an average or a tangent Area divide into small geometrical shapes Day 4 take the simple ideas of reading, slope and area, and through unit analysis, answer any questions on any graph. Day 5 when patterns exist equations can be derived from the graph which eliminates the need of the graph. However, students must understand where these equations came from or how they were derived to prevent their improper use. 7