The Construction and Use of Difference Tables

Air Bag Radial Distance Deployment from Dash Board time distance (ms) inches 10 5 30 15 50 25 Air Bag Radial Distance Deployment from Dash Board Air Bag moves out from the dash board 10 20 30 40 50 5 15 60 70 80 Time (milliseconds) Distance (inches)

Difference Table for the Air Bag Experiment One easy and simple way to make sure you have completed all of the possible calculations needed before the data is analyzed is to just make and then fill in a difference table. Difference Table for the Air Bag Experiment acceleration time distance ms inches 10 5 30 15 50 25 inches/ms 2 0.5 = m velocity 3 1 0 = a

General Format for a Difference Table Difference Tables General Format for a Difference Table Independent variable Dependent variable 1st 2nd 3rd difference difference difference 1 x y 1 1 m 1 x 2 y 2 1 a 2 m 1 c 1 x 3 y 3 2 a 3 m 2 c 1 x 4 y 4 3 a 4 m 3 c 1 x 5 y 5 4 a 5 m 1 x 6 y 6 Note: Most of the time, people only use information from the first 3 difference columns.

Equations for Difference Columns [ y 1 y ] 2 1 m = [ x 1 x 2 ] [ m m ] 2 1 a = 1 3 2 m = y x ] [ [x x ] [ a a ] 2 1 2 1 c = 1 [ x x ] [ m m ] 2 1 3 2 a = 2 [ x x ] 4 3 m = y x ] [ 3 2 [ a a ] 3 2 c = 2 [ x x ] [ m m ] 3 2 4 3 a = 5 4 m = y x ] [ 3 [x x ] 4 2 [ a a ] 4 3 c = [ m m ] 3 5 4 [ x x ] a 4 3 = 6 5 m = y x ] [ 4 [x x ] 5 4 difference 2nd 3rd 1st

Fill in the difference tables. Three Practice Examples dist (m) time (s) 5 0 10 1 15 2 20 3 25 4 30 5 Data, example number 1 dist (m) time (s) 0 0 5 1 15 2 30 3 50 4 75 5 Data, example number 2 dist (m) time (s) 5 0 10 1 20 2 5 3 0 4 10 5 Data, example number 3 Fill in the difference tables.

Difference table calculations for example 1 Plot of Example 1 Data Data for example number 1 1 2 3 4 5 30 10 15 20 25 Time (seconds) Distance (meters) dist (m) time (s) 5 0 10 1 15 2 20 3 25 4 30 5

Difference table calculations for example 1 First Difference Calculations Dependent variable Independent variable dist (m) time (s) 5 0 10 1 15 2 20 3 25 4 30 5 s =m = (d2- d1)/(t2 - t1) m1 = (10 - 5)m/(1 - 0)s = 5m/s m2 = (15 - 10)m/(2- 1)s = 5m/s m3 = (20 - 15)m/(3 - 2)s = 5m/s m4 = etc…

Difference table calculations for example 1 Summary of 1st Difference Calculations speed (m/s) time (s) 5 0 5 1 5 2 5 3 5 4 5 5 Independent variable difference 1st

Fill in the difference table Difference table calculations for example 1 Fill in the difference table 2st 1st dependent variable Independent variable difference difference a = ? 1 2 3 4 m = 5 1 1 y = 5 x = 0 1 m = 5 2 2 y = 10 x = 1 2 a = (s - s ) (t - t ) 2 1 ? = m = 5 3 3 y = 15 x = 2 3 m = 5 4 4 y = 20 x = 3 4 m = 5 5 5 y = 25 x = 4 5 6 y = 30 x = 5 6 acceleration (m/s ) 2 speed (m/s) distance (meters) time (seconds)

Difference table calculations for example 1 Second Difference Calculations a = (s - s ) (t - t ) 2 1 difference 1st Independent variable speed (m/s) time (s) 5 0 5 1 5 2 5 3 5 4 5 5 a1 = (5- 5)(m/s)/(1- 0)s = 0 m/s2 a2 = (5- 5)(m/s)/(2- 1)s = 0 m/s2 a3 = (5- 5)(m/s)/(3- 2)s = 0 m/s2 a4 = etc…

Difference table calculations for example 1 Summary of 2nd Difference Calculations acc (m/s2) time (s) 0 0 0 1 0 2 0 3 0 4 0 5 Independent variable difference 2st

Fill in the difference table Difference table calculations for example 1 Fill in the difference table 2st 1st dependent variable independent variable difference difference 5 5 5 10 1 (s - s ) 5 15 2 2 1 a = 5 20 3 (t - t ) 2 1 5 25 4 5 30 5 acceleration speed (m/s) distance (meters) time (seconds) (m/s ) 2

Data for example number 2 Plot of Example 2 Data 1 2 3 4 5 90 15 30 45 60 75 Time (seconds) Distance (meters) dist (m) time (s) 0 0 5 1 15 2 30 3 50 4 75 5

Difference table calculations for example 2 First Difference Calculations Dependent variable Independent variable s = (d2- d1)/(t2- t1) dist (m) time (s) 0 0 5 1 15 2 30 3 50 4 75 5 s1 = ( 5- 0)m/(1- 0)s = 5m/s s2 = (15- 5)m/(2- 1)s = 10m/s s3 = (30- 15)m/(3- 2)s = 15m/s s4 = etc…

Difference table calculations for example 2 speed (m/s) time (s) 5 0 10 1 15 2 20 3 25 4

Fill in the difference table Difference table calculations for example 2 Fill in the difference table difference 1st Independent variable dependent 2st 1 2 3 4 5 15 30 50 75 difference 1st Independent variable dependent 2st difference 1st Independent variable dependent 2st 5 10 15 20 25 ? 5 5 1 1 15 15 2 2 a = (s - s ) (t - t ) 2 1 ? = 30 30 3 3 50 50 4 4 75 75 5 5 acceleration speed (m/s) distance (meters) time (seconds) (m/s ) 2

Difference table calculations for example 2 Second Difference Calculations a = (s - s ) (t - t ) 2 1 1st Independent variable difference speed (m/s) time (s) 5 0 10 1 15 2 20 3 25 4 a1 = (10 - 5)(m/s)/(1 - 0)s = 5 m/s2 a2 = (15 - 10)(m/s)/(2 - 1)s = 5 m/s2 a3 = (20 - 15)(m/s)/(3 - 2)s = 5 m/s2 a4 = etc…

Fill in the difference table Difference table calculations for example 2 Fill in the difference table 2st 1st dependent variable Independent variable difference difference 5 5 5 10 5 1 5 15 15 2 a = (s - s ) (t - t ) 2 1 5 20 30 3 25 50 4 75 5 acceleration speed (m/s) distance (meters) time (seconds) (m/s ) 2

dist (m) time (s) 5 0 10 1 20 2 5 3 0 4 10 5 Data for example number 3 Plot of Example 3 Data 1 2 3 4 5 30 10 15 20 25 Time (seconds) Distance (meters) Data for example number 3 dist (m) time (s) 5 0 10 1 20 2 5 3 0 4 10 5

Difference table calculations for example 3 First Difference Calculations Dependent variable Independent variable s = (d2- d1)/(t2- t1) dist (m) time (s) 5 0 10 1 20 2 5 3 0 4 10 5 s1 = (10 - 5)m/(1 - 0)s = 5m/s s2 = (20 - 10)m/(2 - 1)s = 10m/s s3 = (5 - 20)m/(3 - 2)s = -15m/s will s4 = a positive or negative number?

Fill in the difference table Difference table calculations for example 3 Fill in the difference table 2st 1st dependent variable Independent variable difference difference 5 5 ? 10 10 1 ? a = (s - s ) (t - t ) 2 1 ? = -15 20 2 ? -5 5 3 ? 10 4 10 5 acceleration speed (m/s) distance (meters) time (seconds) (m/s ) 2

Difference table calculations for example 3 Second Difference Calculations a = (s - s ) (t - t ) 2 1 1st Independent variable difference speed (m/s) time (s) 5 0 10 1 -15 2 -5 3 10 4 a1 = (10- 5)(m/s)/(1- 0)s = 5 m/s2 a2 = (-15- 10)(m/s)/(2- 1)s = -25 m/s2 a3 = (-5- -15)(m/s)/(3- 2)s = 10 m/s2 a4 = etc…

Fill in the difference table Difference table calculations for example 3 Fill in the difference table 2st 1st dependent variable Independent variable difference difference 5 5 5 -25 10 10 1 a = (s - s ) (t - t ) 2 1 ? = 10 -15 20 2 15 -5 5 3 10 4 10 5 acceleration speed (m/s) distance (meters) time (seconds) (m/s ) 2

Summary If all the second difference values are equal to zero, the data will plot as a straight line with the first difference value equal to the slope of the line. If all the second difference values are equal to the same value, the data will plot as the positive half of a parabola. If the second difference value changes sign, the data plot will go in the opposite direction at least once.