Mathematical ModelsChapter 2

By Mr. Leavings

And just what are we going to LEARN?

Construct a speed vs. distance graphUse graphs to make predictionsDetermine the slope of a line (gives you V!)Distinguish between speed and accelerationCalculate acceleration from a formulaCalculate acceleration from the slope of a speed vs. time graph

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?

Scientific Models

Scientific Model: a model that shows how each variable relates to

one another

3 Types:

Physical Models Conceptual Models Graphical Models

Physical Models

We can look, touch, feel and take measurements from them

Often constructed in scale

Conceptual Models

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

Graphical Models

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

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!

Predicting from Graphs

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

Cause and Effect

Strong Relationship

Weak Relationship

Cause and Effect

No Relationship

Inverse Relationship

Position- a comparison from starting point, includes direction.

Distance- an interval of length without regard to direction.

Position and Distance

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

Acceleration

Acceleration = the rate of change in speed of an object

= change in speed

change in time

Acceleration

Acceleration

What units are acceleration in? Lets find out:

Acceleration= change in speed

change in time

Acceleration =

Meters/second___________

Second

Meters/Second2

Acceleration

∆V

Delta V (∆V) is the change in velocity of an object.

∆V = Vf - Vi

Where Vf stands for the final velocity and Vi stands for the initial velocity.

Acceleration

a =

___∆V___ t

OR

a =

__Vf - Vi __ t

Acceleration

a =

__Vf - Vi __ t

Manipulating the equation

then

t =

__Vf - Vi __ a

Vf = Vi + at

and

Vi = Vf - at