Difference Table Construction and Use The Construction and

Difference Table Construction and Use The Construction and Use of Difference Tables 1

Difference Table Construction and Use time distance Air Bag Radial Distance Deployment from Dash Board (ms) 40 Distance (inches) 30 20 15 10 20 30 40 50 60 70 80 Time (milliseconds) Air Bag moves out from the dash board 2 inches 0 0 10 5 30 15 50 25

Difference Table Construction and Use Difference Table 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 time ms distance velocity acceleration inches/ms 0 0 10 5 30 15 50 25 0. 5 = m 1 0. 5 = m 2 0. 5 = m 3 3 0=a 1 2

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

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

Difference Table Construction and Use Three Practice Examples Data, example number 1 dist (m) time (s) 5 0 10 1 15 2 20 3 25 4 30 5 Data, example number 2 dist (m) time (s) 0 0 5 1 15 2 30 3 50 4 75 5 Data, example number 3 dist (m) time (s) 5 0 10 1 20 2 5 3 0 4 10 5 Fill in the difference tables. 6

Difference Table Construction and Use Difference table calculations for example 1 Data for example number 1 Plot of Example 1 Data dist (m) time (s) 5 0 10 1 15 2 20 3 25 4 30 5 Distance (meters) 30 25 20 15 10 5 1 2 3 4 5 Time (seconds) 7

Difference Table Construction and Use 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 = (d 2 - d 1)/(t 2 - t 1) m 1 = (10 - 5)m/(1 - 0)s = 5 m/s m 2 = (15 - 10)m/(2 - 1)s = 5 m/s m 3 = (20 - 15)m/(3 - 2)s = 5 m/s m 4 = etc… 8

Difference Table Construction and Use Difference table calculations for example 1 Summary of 1 st Difference Calculations 1 st difference Independent variable speed (m/s) time (s) 5 5 5 0 1 2 3 4 5 9

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

Difference Table Construction and Use Difference table calculations for example 1 Second Difference Calculations 1 st difference Independent variable speed (m/s) time (s) 5 0 5 1 5 2 5 3 5 4 5 5 a= (s - s ) 2 1 (t - t ) 2 1 a 1 = (5 - 5)(m/s)/(1 - 0)s = 0 m/s 2 a 2 = (5 - 5)(m/s)/(2 - 1)s = 0 m/s 2 a 3 = (5 - 5)(m/s)/(3 - 2)s = 0 m/s 2 a 4 = etc… 11

Difference Table Construction and Use Difference table calculations for example 1 Summary of 2 nd Difference Calculations 2 st difference Independent variable acc (m/s 2) time (s) 0 0 0 1 0 2 0 3 0 4 0 5 12

Difference Table Construction and Use Difference table calculations for example 1 Fill in the difference table 2 st difference 1 st difference dependent variable independent variable 0 0 5 5 5 10 0 1 0 5 15 2 0 5 20 3 0 5 25 4 5 30 5 acceleration (m/s 2 ) speed distance time (m/s) (meters) (seconds) 13 a= (s - s ) 2 1 (t - t ) 2 1

Difference Table Construction and Use Data for example number 2 Plot of Example 2 Data dist (m) time (s) 0 0 5 1 15 2 30 3 50 4 75 5 Distance (meters) 90 75 60 45 30 15 1 2 3 4 5 Time (seconds) 14

Difference Table Construction and Use Difference table calculations for example 2 First Difference Calculations Dependent variable Independent variable dist (m) time (s) 0 0 5 1 15 2 30 3 50 4 75 5 s = (d 2 - d 1)/(t 2 - t 1) s 1 = ( 5 - 0)m/(1 - 0)s = 5 m/s s 2 = (15 - 5)m/(2 - 1)s = 10 m/s s 3 = (30 - 15)m/(3 - 2)s = 15 m/s s 4 = etc… 15

Difference Table Construction and Use Difference table calculations for example 2 speed (m/s) time (s) 5 0 10 1 15 2 20 3 25 4 16

Difference Table Construction and Use Difference table calculations for example 2 Fill in the difference table 2 st 1 st difference ? ? acceleration (m/s 2 ) dependent variable Independent variable 5 10 15 0 5 15 0 1 2 20 30 3 25 50 4 75 5 speed distance time (m/s) (meters) (seconds) 17 ? = a= (s - s ) 2 1 (t - t ) 2 1

Difference Table Construction and Use Difference table calculations for example 2 Second Difference Calculations 1 st difference Independent variable speed (m/s) time (s) 5 0 10 1 15 2 20 3 25 4 a= (s - s ) 2 1 (t - t ) 2 1 a 1 = (10 - 5)(m/s)/(1 - 0)s = 5 m/s 2 a 2 = (15 - 10)(m/s)/(2 - 1)s = 5 m/s 2 a 3 = (20 - 15)(m/s)/(3 - 2)s = 5 m/s 2 a 4 = etc… 18

Difference Table Construction and Use Difference table calculations for example 2 Fill in the difference table 2 st 1 st difference dependent variable Independent variable 5 5 10 15 0 5 15 0 1 2 5 20 30 3 25 50 4 75 5 acceleration (m/s 2 ) speed distance time (m/s) (meters) (seconds) 19 a= (s - s ) 2 1 (t - t ) 2 1

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

Difference Table Construction and Use Difference table calculations for example 3 First Difference Calculations Dependent variable Independent variable s = (d 2 - d 1)/(t 2 - t 1) dist (m) time (s) 5 0 10 1 20 2 5 3 0 4 10 5 s 1 = (10 - 5)m/(1 - 0)s = 5 m/s s 2 = (20 - 10)m/(2 - 1)s = 10 m/s s 3 = (5 - 20)m/(3 - 2)s = -15 m/s will s 4 = a positive or negative number? 21

Difference Table Construction and Use Difference table calculations for example 3 Fill in the difference table 2 st 1 st difference ? ? acceleration (m/s 2) dependent variable Independent variable 5 10 -15 5 10 20 0 1 2 -5 5 3 10 0 4 10 5 speed (m/s) distance (meters) 22 time (seconds) ? =a = (s - s ) 2 1 (t - t ) 2 1

Difference Table Construction and Use Difference table calculations for example 3 Second Difference Calculations 1 st difference Independent variable speed (m/s) time (s) 5 0 10 1 -15 2 -5 3 10 4 a= (s - s ) 2 1 (t - t ) 2 1 a 1 = (10 - 5)(m/s)/(1 - 0)s = 5 m/s 2 a 2 = (-15 - 10)(m/s)/(2 - 1)s = -25 m/s 2 a 3 = (-5 - -15)(m/s)/(3 - 2)s = 10 m/s 2 a 4 = etc… 23

Difference Table Construction and Use Difference table calculations for example 3 Fill in the difference table 2 st 1 st difference dependent variable Independent variable 5 -25 5 10 0 1 10 -15 20 2 15 -5 5 3 10 0 4 10 5 acceleration (m/s 2) speed (m/s) distance (meters) 24 time (seconds) ? =a = (s - s ) 2 1 (t - t ) 2 1

Difference Table Construction and Use 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. 25

Difference Table Construction and Use 26