Objective: Find the proportionality
constant of a directly/inversely
proportional graph.
BASIC PROPORTIONS
Physics is all about. . . patterns
If I change the ___independent variable___ what happens to the __dependent variable__ ?
EXAMPLES: Bob drives his car at a constant speed.
If Bob drives TWICE as fast then his car will move __twice__________ as far.
If Bob drives TWICE as fast it will take him ___four_________ times more distance to stop.
If Bob drives TWICE as fast it will take him ___half_________ as much time to reach the finish line.
When we compare one variable with another it is called a . . .proportion
There are THREE major categories:
1.
DIRECTLY PROPORTIONAL = If the first variable increases the second variable also increases.
2. INVERSELY PROPORTIONAL = If the first variable increases the second variable decreases.
3.
NOT PROPORTIONAL = There is no predictable pattern between the two variables.
DIRECT PROPORTIONS
EQUATION
Y is directly proportional to X.
Y = kX
If X doubles then Y will ___double_____.
How do I solve for k?
Graph Y vs. X
Slope = k
k is called the proportionality constant (every
equation has one and we need to find it).
NATURAL GRAPH
Y
Slope = k
LINEARIZED GRAPH
Don’t Need
X
REMEMBER: In real life . . . graphs are not as perfect as math class
THE PROPER WAY TO CALCULATE SLOPE
Step 1: Make sure the graph is linear.
Step 4: Write both cross points in the form (x,y)
Step 2: Draw a GOOD best fit line.
Step 5: Apply the slope formula to your cross points.
(Slope = rise/run =y2-y1/x2-x1)
Step 3: Circle two good CROSSING points.
Step 6: Write your final slope as a decimal.
You are curious if there is a relationship between the height of a child and its weight. You collect the following data
by measuring all the children on your street.
Weight
(lbs)
31
35
44
51
54
Weight (lbs)
Height
(in)
25
29
33
41
45
Weight vs. Height
100
90
80
70
60
50
40
30
20
10
0
EQUATION (FORM)
Y =kX
Cross Points = (0, 0) (40, 50)
change in y/ change in x
(50-0)/(40-0)=50/40=1.25
K = 1.25
0 5 10 15 20 25 30 35 40 45 50
Height (inches)
REAL WORLD EQN
W = 1.25h
Based on the shape of your graph, the weight of a child is ____directly________ proportional to _____height____.
Use your real world equation to predict how much a child would weigh if she was 52 inches tall.
W=1.25(52) =
65bls
Now try it on your own! Go back to your turtle graph and determine the real world equation.
Now let’s do a real world application. Follow the instructions for the Spring Lab found on the next page.
INVERSE PROPORTIONS
EQUATION
𝐘=
Y is inversely proportional to X.
𝐤
𝐱
If X doubles then Y will ____half________.
How do I solve for k?
𝟏
Graph 𝒀 =
𝒙
& the Slope = k
NATURAL GRAPH
Y
LINEARIZED GRAPH
Y
X
1/x
You are curious if there is a relationship between the time you study for a test and the number of points you miss.
You keep track of your test scores for the year.
How Study Time affects my Grade
Missed (pts.)
Time
1/x
(min)
1/5=.2 5
1/10=.1 10
.04
25
.025
40
.02
50
10
9
Missed 8
(pts) 7
6
10
5
5
4
2
3
1
2
1
1
0
10
9
8
7
6
5
4
3
2
1
0
Linearized Graph
0 5 10 15 20 25 30 35 40 45 50 0 .02 .04 .06 .08 .10 .12 .14 .16 .18 .2
Time (min)
1/x
EQUATION (FORM)
𝑘
𝑌=
𝑥
Cross Points =
(.06, 3) (.08. 4)
(4-3)/(.08-.06)=1/.02=50
K= 50
REAL WORLD EQN
𝟓𝟎
𝐏=
𝐭
Based on the shape of your graph, the points lost on a test are __inversely____ proportional to ___Study Time__.
Use your real world equation to predict how many points you would miss if you only studied for 2 minutes.
Now try it on your own! Go back to your medication graph and determine the real world equation.
Now let’s do a real world application. Follow the instructions for the Mass Lab found on the next page.
P= 50/2
=25 pts.
SPRING LAB
Independent: Distance
Measure the Force (Dependent)
Obj:
Determine the relationship between the stretch distance of a spring and the tension inside the spring.
Design: Place a spring just BEFORE the zero line of a horizontal meter stick.
Pull the spring using an electronic force meter.
Record both the stretch distance and the spring tension.
DC:
Identify the independent & dependent variables. Make a table. Record at least four data points.
DA:
Graph the original data.
If the data is linear, draw a best-fit line. If the data is not linear, draw a second graph to fix it.
From your graph, determine the proportionality constant.
Con:
The tension in a spring is _directly_____ proportional to the __distance stretched__.
The real world equation describing this relationship is This will be the one they find. An ex: T=5d where
T=tension and d=distance.
Eval:
Do you trust your results? Is there anything that went wrong I should be aware of?
MASS LAB
Obj:
Determine the relationship between the mass of a falling weight and the time it takes to move a car.
Design: Attach a wheeled car to a hanging mass using a pulley.
Release the mass and measure the time it takes the car to cross the track.
Add more mass to the hanging weight and repeat.
DC:
Identify the independent & dependent variables. Make a table. Record at least four data points.
DA:
Graph the original data.
If the data is linear, draw a best-fit line. If the data is not linear, draw a second graph to fix it.
From your graph, determine the proportionality constant.
Con:
The time it takes a car to move is _inversely___ proportional to the __pulling mass__.
The real world equation describing this relationship is _This will be the one they find. An ex: t = 3/m
where t = time, and m = mass_.
Eval:
Do you trust your results? Is there anything that went wrong I should be aware of?
The set up for the
lab is shown to
the left.
Note: No photo
gate is needed for
this lab (we were
just lazy and
didn’t take them
off).
The hanger
without bolts is .5
bolt units.
My graphs for this
lab are attached.