Unit 1 Linear Models. Answer. Question. Model. Question. Answer. 3 Types of Models. Model: A simplified version of reality

Transcription

1 Model: A simplified version of reality Model Example: Free Fall is a model of falling objects that ignores air resistance. Free Fall assumes that only gravity affects falling motion. Real fall in an atmosphere means falling through air. Free Fall simplifies this reality. Types of Models 1. Physical models are simpler representations of a thing. Example: A globe is a physical model of the Earth. 2. Conceptual models combine ideas into a story or diagram to explain an event. Example: Space-time is like a fabric that can be curved by masses.. Mathematical models are equations. Example: Distance = Speed x Time

2 When do you include (0,0) as a data point on a graph? Include (0,0) when at X = 0 the Y value is defined as 0 too. Example: You defined that the marble has a position = 0 m at time = 0 s. When you make a scale on a graph axis, what are good division to use on the scale? Example: For a circle of diameter 0 cm, the circumference is 0 cm. (There is no circle.) For decimal data values, choose to mark the scale in units of 1, 2, or 5 (or multiples of 10 of these). For example: 0.1, 0.2, or , 20, , 200, 500

3 Why is 25 a bad division to use on an axis scale? When you read a graph value where divisions are in units of 25, estimating between marks is difficult. It is much easier to estimate between marks when the divisions are by 1, 2, or 5 (or multiples of 10 of these). Why is 0. a bad division to use on an axis scale? When you read a graph value where divisions are in units of 0., estimating between marks is difficult. It is much easier to estimate between marks when the divisions are by 1, 2, or 5 (or multiples of 10 of these).

4 Linear Data looks like a line when you graph the data. Linear Data Heart Rate (BPM) Exercise Time (hr/week) Linear Model Linear Model is an equation of a best fit line, drawn through linear data. Heart Rate (BPM) H = (-4 BPB)t + (79 BPM) H Resting Heart Rate (BPM)H TT Exercise Time (hr/week) Exercise Time (hr/week)

5 1. Graph the data. x ( cm) How do you find a linear model from a graph? Time (s) 2. Draw a best fit line through the data. x ( cm) Time (s) y y. Find the slope with units: Slope = x x 4. Find the y-intercept with units. 5. Plug into linear equation: Y = mx + b

6 How do you decide which line is the best fit line? Volume (cm^) Volume vs. Temperature of a Gas Bad Temperature ( C) Good Bad Half the points are above the line, and half the points are below the line Choose smallest distance between the points and the line (vertically) Do not force the line to go through the data connect the dots

7 How do you find the slope of a line? 1. Mark and label any 2 points on a line that are easy to read. These points don t have to be data points. 2. Use the equation: Include units. y Slope = x y x Slope = = = = y2 y1 x2 x1 00cm 250cm 80 C 20 C 50cm 60 C 0.8 cm C

8 Volume (cm^) How do you find the y-intercept of a line? Volume vs. Temperature of a Gas Temperature ( C) Y intercept tells us the Y value when X = 0 On the graph, this is the Y value where the line crosses the Y axis In this example: When Temperature is 0⁰C, the Volume is 2 cm. So, y-intercept = 2 cm

9 Volume (cm^) How do you plug into y = mx + b to find the linear model? Volume vs. Temperature of a Gas Temperature ( C) X variable: Temperature Call it T Y variable: Volume Call it V cm Slope = 0.8 C y-intercept is V = 2 cm Plug everything into cm V = 0.8 2cm C T + ( )

10 How would you use this model to predict the volume of a gas at a temperature of 100 ⁰C? cm V = 0.8 2cm C T + How would you use this model to predict the distance D after t = 4 hours? D 4 mi = t h ( ) Math models are useful for predicting values you don t know. To make a prediction, plug in the temperature value: cm V = 0.8 T + ( 2cm ) C cm = C C + 2cm = 8+ 2 = 16cm ( ) ( ) Math models are useful for predicting values you don t know. Here we must assume that the object travels at constant speed for these 4 hours, or the model doesn t work! To make a prediction, plug in the time value: mi mi D = 4 t = 4 ( 4h) = 16mi h h

11 Interpret this slope in words: cm 0.8 C cm 0.8cm 0.8 = C 1C For each extra 1⁰C, the volume increases 0.8 cm. Interpret this slope in words: 4 m s m 4 = s 4m 1s For each extra 1 s, the position decreases 4m.

12 From a Position vs. Time graph, interpret this y-intercept in words: 79 m On a Position vs. Time graph, Position is on the Y-axis and Time is on the X-axis. At 0 s, the position is 79 m. From a Velocity vs. Time graph, interpret this y-intercept in words: - 45 m/s On a Velocity vs. Time graph, Velocity is on the Y-axis and Time is on the X-axis. At 0 s, the velocity is -45 m/s.

13 What is Nonlinear Data? Nonlinear data looks curved on a graph. Would a linear model be a good fit for this data? Why or why not? No. The data is curved and a line is not a good representation of this data. This is an example of nonlinear data.

14 Why are all models limited in their usefulness for making predictions? A model is a simplified version of reality, which may predict nonsense outside its data range. This model predicts 0 and then negative Resting Heart Rate (death) if the exercise time is more than 20 h/wk, which is false. Heart Rate (BPM) Why is this model limited in prediction distance traveled? Distance = Speed x Time Exercise Time (hr/week) This model requires that the object move at constant speed, or that you know its average speed. If the speed is NOT constant and you don t know the average speed, this model won t correctly predict distance traveled.