Graph Manipulation

The Goal of Graphing

• Create a graph of a linear plot with a positive slope. Use a linear line of best fit to see how close your data is to actually being linear.

• This type of graph helps us to quickly see the RELATIONSHIP BETWEEN VARIABLES, NOT how variables change.

Linear Form• All relationships can be fit to “y=mx+b”• y = dependent variable• m = slope gradient• x = independent variable• b = y-intercept

Two Approaches:

• Equation First (you know the equation)

• Data First (you don’t know the equation, but can use data to create one!)

• Let’s start with something familiar: An object dropped from rest fits: s=½at2

Equation First: • How does the acceleration due to gravity on

different planets affect the displacement of a ball two seconds after being dropped?

• Independent:• Dependent:• Gradient:

• Rearrange equation if needed…

€

s = 12at2

Location Acceleration / ms-2 Displacement / mMoon 1.64 3.3Venus 8.83 17.6Earth 9.81 19.6Saturn 10.44 21.2Neptune 14.07 28.2

€

Remember m= 12t2 = 1

222 = 2 ≈2.0076

Equation First: • How does the time a ball falls affect its

displacement?

• Independent:• Dependent:• Gradient:

• Rearrange equation if needed…

€

s = 12at2

Time / s Displacement / m0.5 1.231 4.91

1.5 11.042 19.62

2.5 30.663 44.15

Time / s Displacement / m Time2 / s2

0.5 1.23 0.251 4.91 1

1.5 11.04 2.252 19.62 4

2.5 30.66 6.253 44.15 9

Equation First: • How does the distance a ball falls affect the

time it spends in the air?

• Independent:• Dependent:• Gradient:

• Rearrange equation if needed…

€

s = 12at2

Displacement / m Time / s1 0.452 0.643 0.784 0.95 1.016 1.11

Displacement / m Time / s √Displacement / m.5

1 0.45 12 0.64 1.413 0.78 1.734 0.9 25 1.01 2.246 1.11 2.45

Equation First: • How does the acceleration due to gravity

affect the time it takes a ball to drop 3 m?

• Independent:• Dependent:• Gradient:

• Rearrange equation if needed…

€

s = 12at2

Location Acceleration / ms-2 Time / sMars 3.77 1.26

Venus 8.83 0.82Earth 9.81 0.78

Saturn 10.44 0.76Neptune 14.07 0.65

Location Acceleration / ms-2 Time / s 1/Acceleration / m-1s2

Mars 3.77 1.26 0.265Venus 8.83 0.82 0.113Earth 9.81 0.78 0.102

Saturn 10.44 0.76 0.096Neptune 14.07 0.65 0.071

Location Acceleration / ms-2 Time / s 1/Acceleration / m-1s2

Mars 3.77 1.26 0.265Venus 8.83 0.82 0.113Earth 9.81 0.78 0.102

Saturn 10.44 0.76 0.096Neptune 14.07 0.65 0.071

Location Acceleration / ms-2 Time / s 1/Acceleration / m-1s2

Mars 3.77 1.26 0.265Venus 8.83 0.82 0.113Earth 9.81 0.78 0.102

Saturn 10.44 0.76 0.096Neptune 14.07 0.65 0.071

Location Acceleration / ms-2 Time / s 1/Acceleration / m-1s2 √1/Acceleration / m-.5sMars 3.77 1.26 0.265 0.52

Venus 8.83 0.82 0.113 0.34Earth 9.81 0.78 0.102 0.32

Saturn 10.44 0.76 0.096 0.31Neptune 14.07 0.65 0.071 0.27

(BREAK)

Data First: • How does the time a ball falls affect its

displacement (when g=9.81 ms-2)

Time / s Displacement / m0.5 1.231 4.91

1.5 11.042 19.62

2.5 30.663 44.15

Time / s Displacement / m Time2 / s2

0.5 1.23 0.251 4.91 1

1.5 11.04 2.252 19.62 4

2.5 30.66 6.253 44.15 9

Time / s Displacement / m Time2 / s2

0.5 1.23 0.251 4.91 1

1.5 11.04 2.252 19.62 4

2.5 30.66 6.253 44.15 9

€

y =4 .905 ⋅x + 0 .0034

€

s =4 .905 ⋅t2 + 0 .0034

€

s =4 .905 ⋅t2

€

s = 12a⋅t2

€

y =4 .905 ⋅x + 0 .0034

€

s =4 .905 ⋅t2 + 0 .0034

€

s =4 .905 ⋅t2

€

s = 12a⋅t2

YOU JUST DISCOVED A FUNDAMENTAL EQUATION OF

PHYSICS SIMPLY BY MANIPUALTING AND ANALYZING A GRAPH.

A string is tied to a 2.0 kg mass and swung in a circle at a constant 3.0 ms-1. The length of the string is changed and the force is measured.

Thirsty For More?

Length / m Force / N0.2 950.4 470.6 320.8 25

1 19

Length / m Force / N 1/Length / m-1

0.2 95 50.4 47 2.50.6 32 1.670.8 25 1.25

1 19 1

1/Length

• A string is tied to a 2.0 kg mass and swung in a circle at a constant 3.0 ms-1. The length of the string is changed and the force is measured.

€

y =18 .053 ⋅x + 0 .7677

€

F =18 .053 ⋅1L+ 0 .7677

€

F =18 .053 ⋅1L

€

F =m⋅v2 ⋅1L

€

→ F =mv2

L

€

y =18 .053 ⋅x + 0 .7677

€

F =18 .053 ⋅1L+ 0 .7677

€

F =18 .053 ⋅1L

€

F =m⋅v2 ⋅1L

€

→ F =mv2

L

€

!YOU DID IT AGAIN !!

Fc =mv2

r