1. The Construction and Use of Difference Tables

2. 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)

3. 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

4. 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.

5. 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

6. 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.

7. 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

8. 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 = ( )m/(2- 1)s = 5m/s
m3 = ( )m/(3 - 2)s = 5m/s
m4 = etc…

9. 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

10. 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)

11. 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…

12. 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

13. 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

14. 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

15. 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 = ( )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…

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

17. 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

18. 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)
a1 = (10 - 5)(m/s)/(1 - 0)s = 5 m/s2
a2 = ( )(m/s)/(2 - 1)s = 5 m/s2
a3 = ( )(m/s)/(3 - 2)s = 5 m/s2
a4 = etc…

19. 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

20. 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

21. 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 = ( )m/(2 - 1)s = 10m/s
s3 = (5 - 20)m/(3 - 2)s = -15m/s
will s4 = a positive or
negative number?

22. 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

23. 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)
a1 = (10- 5)(m/s)/(1- 0)s = 5 m/s2
a2 = ( )(m/s)/(2- 1)s = -25 m/s2
a3 = ( )(m/s)/(3- 2)s = 10 m/s2
a4 = etc…

24. 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

25. 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.