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Mathematical Models Chapter 2 By Mr. Leavings

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And just what are we going to LEARN? Construct a speed vs. distance graph Use graphs to make predictions Determine the slope of a line (gives you V!) Distinguish between speed and acceleration Calculate acceleration from a formula Calculate acceleration from the slope of a speed vs. time graph

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Mathematical Models Why would you want to make a model? To answer complicated questions it is easier to break down the problem into more manageable pieces. Example from your reading: Building a train. How powerful of a motor do we need? How strong of brakes to stop the train? How much fuel to travel the distance required?

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Scientific Models Scientific Model: a model that shows how each variable relates to one another 3 Types: Physical Models Conceptual Models Graphical Models

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Physical Models We can look, touch, feel and take measurements from them Often constructed in scale

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Conceptual Models These types of models are descriptive. We use them to describe how something works.

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Graphical Models Graphical Models: use graphs to show the relationship between the variable on the x axis and the variable on the y axis.

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Graphical Models Dependent Variable: the measurement that changes based on the independent variable. Also the data that we measure. Independent Variable: the measurement that we change to determine its effect on the dependent variable. Independent variable ALWAYS placed on the x axis! Dependent Variable is ALWAYS placed on the y axis!

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Predicting from Graphs The purpose of making a graph is to organize your data into a model so that you can make predictions.

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Cause and Effect Strong Relationship Weak Relationship

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Cause and Effect No Relationship Inverse Relationship

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Position- a comparison from starting point, includes direction. Distance- an interval of length without regard to direction. Position and Distance

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Slope is the ratio of “rise” (vertical change) to the “run” (horizontal change) of a line. ◦ The rise is determined by finding the height of the triangle shown. ◦ The run is determined by finding the length along the base of the triangle. Determining Speed

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Acceleration

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Acceleration = the rate of change in speed of an object = change in speed change in time Acceleration

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What units are acceleration in? Lets find out: Acceleration = change in speed change in time Acceleration = Meters/second ___________ Second Meters/Second 2

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Acceleration ∆V Delta V (∆V) is the change in velocity of an object. ∆V = V f - V i Where V f stands for the final velocity and V i stands for the initial velocity.

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Acceleration a = ___∆V___ t OR a = __V f - V i __ t

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Acceleration a = __V f - V i __ t Manipulating the equation then t = __V f - V i __ a V f =V i + at and V i =V f - at

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